HUE  on  the  last  date  stamped  below 


UNj\/-'r  "   BRANCH 

CALlFORNfA 
L/BRARY 

L-'S    ANGELES.  CAUF. 


FREEHAND    PERSPECTIVE 
AND  SKETCHING 


FREEHAND 
PERSPECTIVE 
AND    SKETCHING 

PRINCIPLES  AND  METHODS  OF 
EXPRESSION  IN  THE  PICTORIAL 
REPRESENTATION  OF  COMMON 
OBJECTS,  INTERIORS.  BUILDINGS 
AND    LANDSCAPES 

BY 

DORA    MIRIAM    NORTON 

INSTRUCTOR     IN     PERSPECTIVE,     SKETCHING 
AND    COLOR,    PRATT    INSTITUTE,    BROOKLYN 

FOURTH    EDITION 


BROOKLYN 
PUBLISHED   BY   THE   AUTHOR 

1916 

0  S9  18 


N 

Copyright,  1908 

By  Doua  Miriam  Norton 


C139 


THE   UNIVERSITY    PRESS,    CAMBRIDGE,    V.  S.  A. 


TO  THE 

MEMORY   OF   WALTER   SMITH 

FIRST    DIRECTOR    OF    THE    MASSACHUSE'lTS    NORMAL    ART    SCHOOL 
INSPIRING    CRITIC    AND    JUDICIOUS    FRIEND 

THIS   BOOK  IS  DEDICATED 

WITH    THE    WISH    THAT    IT    MAY    HELP    OTHERS    AS    ITS 
AUTHOR    HAS    BEEN    HELPED 

D.  M.  N. 


iz/rs' 


PREFACE 

REVISED   FOR   THE   SECOND   EDITION 

THIS  book  presents  essentially  the  course  of  study  in  Free- 
hand Perspective  and  Sketching  as  developed  during  its 
teaching  at  Pratt  Institute  since  the  founding  of  the 
institute  in  1887.  It  consists  of  a  series  of  illustrated  exercises 
with  explanatory  text,  so  covering  the  subject  that  students  who 
follow  the  course  as  directed  acquire  the  power  to  draw  with  ease 
and  intelligence,  not  only  from  objects,  but  from  memory  and 
from  descriptions.  The  principles  and  methods  here  set  forth 
have  been  taught  by  the  author  for  some  years  in  the  above  school, 
and  have  been  found  practically  effective  in  that  direction.  In 
revising  it  the  author  has  drawn  upon  its  use  as  a  text-book 
for  large  classes,  as  well  as  in  other  directions. 

As  offered  to  the  public  this  course  is  intended  to  form  a  text- 
book for  classes  in  high,  normal,  and  technical  schools  and  in 
colleges ;  also  as  a  book  of  reference  for  supervisors  and  teachers 
of  drawing,  for  draughtsmen  and  artists  whose  training  in  per- 
spective needs  to  be  supplemented,  and  for  the  instruction  of 
students  so  situated  that  personal  art  teaching  is  beyond  their 
reach.  Since  manuals  for  the  teaching  of  drawing  to  children 
already  exist,  the  methods  here  presented  are  primarily  such  as 
have  been  found  effective  with  maturer  minds.  Its  relation 
to  the  teaching  of  children  is  thus  like  that  of  a  grammar  to 
the  **  language  lessons ' '  of  the  primary  schools.  From  it  the 
teacher,  whether  of  children  or  of  adults,  may  select  material  for 
courses  according  to  age  or  aims  in   study. 

In  the  case  of  older  students,  though  perspective  books  excellent 
in  certain  directions  have  been  published,  it  has  been  found  diffi- 
cult to  direct  inquirers  to  anything  at  once  applicable  to  immedi- 
ate  use  and  comprehensive  enough  to  give  a  working  knowledge  of 


PREFACE 

the  subject.  For  several  years,  therefore,  the  need  which  this 
book  is  intended  to  meet  has  been  increasingly  felt.  In  the  hope 
that  it  may  pass  on  to  others  the  aid  received  in  the  past  it  is 
sent  forth. 

The  author  gladly  acknowledges  indebtedness  to  many  sources 
in  the  making  of  this  volume.  Although  the  naming  of  all  would 
be  impossible  in  this  brief  space  some  are  so  preeminent  that 
mention  cannot  be  forborne.  The  experiences  of  teaching  the 
subject  under  the  care  of  Mr.  Walter  Smith,  then  Director  of 
the  Massachusetts  Normal  Art  School,  and  later  with  Mr.  Walter 
Scott  Perry,  Director  of  the  School  of  Fine  and  Applied  Arts  of 
Pratt  Institute,  himself  a  teacher  of  great  originality  and  force  in 
the  subject  for  some  years,  have  been  most  fruitful  in  the  accumu- 
lation of  subject  matter  for  this  course.  The  author's  drawings 
have  been  so  largely  and  sympathetically  supplemented  by  the 
work  of  Mr.  Ernest  W.  Watson,  now  teaching  in  the  Institute, 
as  to  merit  appreciation  beyond  that  due  the  ordinary  illustrator. 
In  the  bringing  out  of  the  book  the  practical  advice  of  Mr. 
C.  Franklin  Edminster,  many  years  an  instructor  in  the  same 
school,  and  the  critical  taste  of  Mr.  Henry  Lewis  Johnson,  editor 
of  Tlie  Printing  Art,  to  whose  suggestion  is  owing  the  form  in 
which  the  book  appears,  have  been  of  great  value.  Of  these  aids, 
and  of  others  not  mentioned,  it  is  a  pleasure  to  here  express  a 
grateful  appreciation. 

D.  M.  N. 

Brooklyn,  June  24,  1910. 


CONTENTS 


Page 

Introduction xi 

Chapter 

I.     General  Directions 1 

II.     Pencil  Measurement  and  the  Picture  Plane 4 

III.  The  Ellipse 8 

IV.  A   Cn-INDER   AND   A   CYLINDRICAL   ObJECT 12 

V.     An  Object  above  the  Eye  and  the  Cone  Principle 18 

VI.     A  Cream  Jug 20 

VII.    A  Time  Study 24 

VIII.     A  Group  of  Cylindrical  Objects 26 

IX.     Cylindrical  Objects  jGrouped  with  Fruit 29 

X.    A  Group  of  OsjECTS^-wtOM  Memory  or  Invention 31 

XI.    The  Cylinder  Cone  and  Ball  Grouped  —  A  Problem  for  Original^ 

Study 34 

XII.     The  Study  of  Straight  Line  Objects 36 

XIII.  Drawing  the  Book  in  Two  Positions 43 

XIV.  The  Book  with  a  Cylindrical  Object 45 

XV.     The  Cylinder  and  Rectangular  Block  ^  A  Problem  for  Original 

Study 48 

XVI.     The  Further  Study  of  Straight-Line  Objects  —  A  Cube  at  Angles 

WITH  the  Picture  Plane 49 

XVII.     The  Cube  in  Two  Different  Positions 53 

XVIII.     A  Book  at  Angles  to  the  Picture  Plane 58 

\    XIX.     Two  Books  at  Different  Angles  to  the  Picture  Plane   ....  61 
-^XX.    The  Actual  Center  of  the   Circle  and   Measurement   into   the 

Picture  by  Parallel  Lines 63 

XXL     Books  with  a  Cylindrical  Object 67 

XXII.     The  Study  and  Drawing  of  a  House       69 

XXIII.  A  Building  from  the  Photograph  or  a  Print 81 

XXIV.  Type  Forms  Helpful  in  Understanding  the  House  — The  Square 

Frame 85 

XXV.     The    Square  Pyramid  and  Square  Plinth 88 

XXVI.     The  Square  Frame  Leaning  on  the  Rectangular  Block  —  A  Prob- 
lem FOR  Original  Study 9j 

ix 


CONTENTS 

Chapter  Page 

XXVII.     Cylindrical  Objects  when  not  Vertical 92 

XXVIII.     A  Group  of  Flower  Pots 95 

XXIX.     The  Circular  Frame  in  a  Square  Frame 96 

XXX.     A  Round  Window 100 

XXXI.     The  Clock  —  A  Problem 102 

XXXII.     The  Arch • 103 

XXXIII.  Interiors  — A  Room  Parallel  to  the  Picture  Plane      ....  105 

XXXIV.  Interiors  Continued— A  Room  at  Angles  to  the  Picture  Plane  110 
XXXV.     Further  Studies  of  Interiors 114 

XXXVI.    A  Chair 118 

XXXVII.  The  Hexagonal  Plinth  in  Two  Positions 121 

XXXVIII.  Interior  with  a  Tiled  Floor 126 

XXXIX.     The  Hexagonal  Prism  and  Frame 128 

XL.     The  Triangular  Prism  and  Frame  —  A  Problem  for  Original 

Study 131 

XLI.     The  Study  op  Parallel  Perspective 132 

XLII.     A  Street  from  the  Photograph 137 

XLIII.     Exceptions  to  the  Use  of  the  Flat  Picture  Plane 139 

XLIV.     Shadows 143 

XLV.    Out-of-doors  Work 154 

SOLUTIONS  OF  PROBLEMS  , I6I 

INDEX 171 


INTRODUCTION 

FREEHAND  Perspective  teaches  those  few  principles 
or  truths  which  govern  the  appearance  of  things  to 
the  eye,  and  the  application  of  these  principles  to  the 
varied  conditions  encountered  in  drawing.  Strictly  speaking, 
there  are  but  two  foundation  truths  in  perspective,  namely: 

First.  Things  appear  smaller  in  proportion  to  their  dis- 
tance from  the  eye.  A  house  ten  rods  distant  can  be 
wholly  seen  through  one  pane  of  glass  (Fig.  8, 
Ch.  II). 

Second.  The  eye  can  see  surfaces  in  their  true 
shape  only  tvhen  placed  at  7'ight  angles  to  the  direc- 
tion in  which  the  eye  looks,  or,  generally  speaking, 
parallel  to  the  face.  When  not  so  placed  they  ap- 
pear lessened  in  one  dimension,  that  is,  either  nar- 
rowed or  shortened,  in  proportion  as  they  are 
turned  away  from  the  face  or  tend  to  coincide 
with  the  direction  of  seeing.  This  apparent  change  of  shape  is 
Foreshortening.  The  cylinder  top  held  at  right  angles  to  the 
direction  of  seeing  appears  as  a  circle  (A  in  Fig.  1).  When 
turned   away   from   this   direction   (as   at   B),   it    appears   nar- 


-^ 


n  rowed,  or  foreshortened.     So  the   pencil  seen   its 

/'^'-^'^    full   length   at  A  in   Fig.  2  appears  foreshortened 

/y\    y\    when  held  as  in  B.     All  the  phenomena  of  free- 

'    A  'I      B     hand  perspective,  however  complicated   and  per- 

^^^'  ^  plexing,   may  be   simplified    by  referring   to  one 

or  both  of  these  principles. 

One    great    obstacle    to   the    ready  mastery  of    these    prin- 
ciples is  our  knowledge   of  the  actual  shapes  of  objects.     For 

xi 


FREEHAND     PERSPECTIVE 

instance,  we  hnoiv  the  top  of  a  cylinder  (B,  Fig.  1)  to  be 
in  fact  a  circle,  and  therefore  we  tend  to  mentally  see  a  circle, 
though  it  is  just  as  truly  a  fact  that  the  top  can  only  appear 
to  the  eye  as  a  circle  when  the  cylinder  is  held  so  as  to  lose 
sight  of  all  other  parts  of  it,  as  at  A.  Consequently,  the  first 
aim  and  benefit  in  studying  perspective  is  the  learning  to  see; 
that  is,  to  know  what  is  the  image  really  presented  to  the  eye. 
Therefore  no  step  should  ever  be  passed  without  clearly  see- 
ing the  appearance  under  consideration.  And  in  all  drawings 
the  final  test  must  be  the  eye;  for,  unless  the  drawing  looks 
right,  it  is  not  right.  All  rules  and  tests  are  only  means  to 
this  end. 

Furthermore,  the  right  study  of  perspective,  which  is  think- 
ing and  drawing  in  perfect  coordination,  enables  the  student 
to  draw  objects  singly  or  combined  or  in  unfamiliar  positions, 
without  having  them  in  sight.  Also  he  should  be  able  to 
draw  an  object  which  he  has  never  seen  if  a  description  of  it 
can  be  supplied.  That  this  last  is  quite  possible  any  prac- 
tical artist  will  agree.  The  writer  recalls  hearing  a  popular 
illustrator  ask  in  a  company  of  friends,  "  Does  any  one  know 
what  a  cider  press  is  like?  "  adding  that  he  must  put  one 
in  an  illustration  with  no  chance  to  see  the  thing  itself.  No 
doubt  of  the  sufficiency  of  a  description  was  expressed,  and  in 
this  case  it  must  suffice  —  a  not  uncommon  situation.  Hence 
the  necessity  of  memory  work  and  dictation  problems,  such  as 
form  part  of  this  course  of  study. 

Finally,  it  is  not  intended  that  in  later  practical  work  drawings 
should  be  actually  constructed  by  the  explanatory  methods  here 
given.  These  exercises  should  be  drawn  as  directed,  since  only 
by  the  actual  experience  of  doing  it  can  their  principles  be  mas- 
tered,, but  a  rigid  clinging  to  these  methods  in  practice  would 
result  in  very  little  art.  Freehand  Sketching  means  dratving  by 
the  trained  eye  and  judgment^  only  using  constructive  methods  to 
test  new  or  doubtful  points.  It  is  to  make  such  sketching  valu- 
able by  a  foundation  of  definite  knowledge  that  these  methods 

xii 


INTRODUCTION 

are  given.  The  trained  artist  draws  a  vase  in  his  flower  study,  or 
a  round  tower  in  a  landscape  with  no  distinct  recalling  of  ellipse 
laws,  feeling  only  joy  in  the  living  curves  as  they  spring  out 
under  his  hand.  But  he  would  labor  long  and  wearily  over  their 
shaping  had  he  not  this  foundation  knowledge,  which  he  uses 
almost  unconsciously. 


xiu 


Chapter  I 


GENERAL   DIRECTIONS 


MATERIALS.  —  Any  paper  having  a  fine  and  fairly  soft 
texture  can  be  used.  It  should  produce  an  even 
grain  in  both  vertical  and  horizontal  pencil  strokes. 
Pencil  exercises  such  as  those  reproduced  in  this  book  are 
usually  drawn  on  paper  of  quarter  imperial  size  (11"  x  15"), 
on  which  at  least  an  inch  and  a  half  of  margin  is  allowed. 
This  is  a  good  size  for  the  student's  drawings,  whether  copied 
from  these  exercises  or  drawn  from  objects.  Have  two 
pencils,  one  fairly  soft  (as  No.  2  Faber,  SM  Dixon,  or 
2  B  Koh-i-noor) ,  and  a  harder  one ;  also  a  good  eraser. 

Line  Practice.  —  Cut   the   pencil   like   the    illustration 
(Fig.  3),  and  rub  on  practice  paper  ^  till  a  broad  line, 

firm  at  the  edges 
and  transparent 
(that  is,  with  the 
grain  of  the  paper 
slightly  showing 
through  it)  can  be 
made.  Sit  erect, 
with  the  paper  directly 
in  front,  and  have  the 
desk  top  inclined,  or  use 
a  drawing  board  (Fig.  4), 
that  the  paper  may  be  as 
nearly  as  possible  parallel  with  the  face.  Hold  the  pencil  almost 
flat,  as  in  the  illustration  (Fig.  5),  and  as  loosely  as  is  consistent 

^  Save  spoiled  sheets  for  this.     Practice  paper  should  be  like  that  on  which  drawings  are 
made. 


Fig.  3 


Fig.  4 


FREEHAND    PERSPECTIVE 


Fig. 


with  a  steady  control.  For  horizontal  lines  use  position  A, 
Fig.  5,  moving  the  pencil  from  left  to  right;  for  vertical  lines 
use  position  B,  moving  from  the  top  downward.  Practice 
vertical,  horizontal,  or  oblique  lines  persistently;  moving  the 
hand  freely  from   the   shoulder,  not  resting  it  on  the  wrist   or 

elbow.  If  the  muscles  acquire  an 
unpleasant  tension,  relax  by  dropping 
the  hands  at  the  sides  and  loosely 
shaking  them.  Unfamiliar  or  diffi- 
cult exercises  should  be  first  carefully 
sketched  with  a  thin,  light  line.  If 
wrong,  draw  over  without  erasing 
until  a  satisfactory  form  is  obtained. 
Erase  the  incorrect  part,  and  ren- 
der expressively  (Ch.  IV).  But  after 
the  composition  of  the  exercise  is 
planned,  such  straight  lines  as  mar- 
gins, cylinder  sides,  and  many  ellipses  may  be  drawn  in  full  at 
once.  And  as  the  student  gains  in  skill,  more  and  more  of  the 
work  should  at  the  first  touch  be  put  on  the  paper  as  it  is 
intended  to  remain.  Exact  knowledge  is  to  be  acquired  only 
that  artistic  interpretations  may  be  expressed  with  ease  and 
certainty. 

Models  for  "Work.  —  Objects  in  common  use  have  been  chosen 
for  most  of  these  exercises.  Geometric  solids  are  assigned  only  as 
needed  for  the  clearer  elucidation  of  perspective  truths.  Neces- 
sary models,  as  the  cylinder,  the  cube,  and  others,  should  be  made 
by  the  student  as  directed.  For  forms  (as  the  hexagonal  frame) 
too  complicated  to  be  easily  made,  the  well-known  wooden 
models  have  been  used.  But  after  thorough  mastery  of  the 
simpler  forms,  most  of  the  later  lessons  can  be  understood  with- 
out models. 

Placing  of  Models.  —  All  objects  for  study  should  be  placed  so 
as  to  present  their  vertical  surfaces  in  nearly  their  true  shape  to 
the  student.     Thus  if  the  model  is  to  be  near,  as  on  the  table 

2 


GENERAL    DIRECTIONS 


at  which  the  student  sits,  it  is  better  to  raise  it  a  few  inches 
(Fig.  4).  This  will  not  be  necessary  if  it  can  be  placed  four  or 
five  feet  distant.  If  the  study  is  seen  too  much  from  the  top, 
the  perspective  will  be  unpleasantly  violent,  as  in  a  photograph 
where  the  camera  has  been  pointed  too  much  downward. 

The  Table  Line.  — To  indicate  a  supporting  surface  under  the 
objects  a  horizontal  line  (A,  B  in  Fig.  6)  is  used.  It  stands  for 
the  back  edge  of  the  table  or  other  horizontal  support- 
ing surface,  and  is  called  the  Table  Line.  It  should  be 
represented  as  further  back  than  any  portion  of  the 
study.  As  will  be  observed  later,  it  need  not  be  used 
if  the  supporting  surface  is  otherwise  suggested,  as  by 
a  cast  shadow  (Fig.  34). 

All  Work  Freehand.  — All  work  is  to  be  done  freehand,  that 
is,  with  no  ruling,  and  no  measuring  other  than  by  the  eye 
and  pencil. 


Fig.  6 


Chapter  II 


PENCIL  MEASUREMENT  AND  THE 
PICTURE  PLANE 

PENCIL  Measurement.  —  Before  studying  the  exercises  which 
follow,  the  beginner  should  become  familiar  with  Pencil 
Measurement.  Place  a  book  upright  directly  in  front  of 
the  eye.  With  one  eye  shut  and  the  arm  at  full  length  (to  ensure 
a  uniform  distance  from  the  eye)  measure  on  the  pencil  held  hori- 
zontally the  apparent  width  of  the  book.  Then  turning  the  pen- 
cil, compare  this  dis- 
tance with  its  height 
(Fig.  7).  (It  is  bet- 
ter to  take  the  smaller 
distance  first,  and  to 
measure  it  into  the 
larger.)  Compare  the 
proportions  so  found 
with  those  obtained 
by  actual  measure- 
ment of  the  book. 
But  always  get  the 
pencil  measurement 
first,  for  this  compels  the  eye  to  do  all  that  it  can  unaided 
before  showing  by  actual  measurement  how  much  better  it  can 
learn  to  do. 

Now  turn  the  book  away  a  little,  and  compare  this  new  ap- 
pearance of  the  width  with  the  height  (Fig.  12). 

The  Picture  Plane.  —  Here  we  must  learn  to  keep  tlie  pencil 
parallel  tvith  the  face  in  order  that  the  pencil  measurement 
may  be  reliable.     For  this,  go  to  the  window,  and  stand  facing 

4 


Fig.  7 


PENCIL    MEASUREMENT,    ETC. 


WINDOV(/   USED  AS 


PCCTt/BE 

,  Plane 

z 

o 

P 

e 

a 

J 

•0 

a 

2 

<J 

B.  Plan  of  A 


A.  Showing  use 

OF   WINDOW   AS 
PICTURE    PUANE 


the  glass,  so  the  face  is  parallel  with  it.  Choose  some  object 
seen  through  the  window,  as  another  house,  and  resting  the 
pencil  against  the  glass  measure  its  width  and  compare  that 
with  its  height 
(Fig.  8). 

Observe  that 
if  the  outline  of 
the  house  could 
be  traced  by  the 
pencil  on  the 
glass  it  would 
form  correctly 
the  apparent 
shape  of  that 
house. 

This  leads  us 
toseethata//j!;fr- 
spective  drcming 
may  be  regarded 
as  placing  on 
paper  the  equiva- 
lent of  such  a  tracing  on  the  glass.  It  will  therefore  be  apparent 
at  once  that  pencil  measurement,  to  be  correct,  must  be  taken 
with  the  pencil  held  as  if  laid  on  such  a  pane  of  glass;  or  in 

other  words,  07i  a  plane  parallel  ivith  and  in 
front  of  the  face.  This  imaginary  transparent 
plane  is  called  the  Picture  Plane,  and  is  a 
most  important  factor  in  all  freehand  draw- 
ing. Thus,  by  turning  or  revolving  the 
pencil  on  the  glass  in  front  of  the  face, 
that  is,  by  revolving  the  pencil  in  the  picture 
plane,  it  can  he  made  to  cover  the  appearance  of  any  possible  line  or 
direction.  For  example,  the  sloping  gable  edge  of  the  outside 
house,  though  retreating  from  the  eye  and  therefore  foreshort- 
ened, can  be  covered  by  the  revolving  pencil  (Fig.  9),  thus  giving 

5 


Fig.  8 


.•iflHH 


Fig.  9 


FREEHAND    PERSPECTIVE 

the  appearance  or  picture  of  its  direction.  Its  apparent  or  fore- 
shortened length  can  also  be  taken  on  the  pencil  and  compared 
with  any  other  dimension,  as  the  height  of  the  nearest  corner. 
The  essential  requirement  is  that  the  pencil  shall  constantly  lie  flat 
on  this  pane  of  glass;  that  is,  on  the  picture  plane. 

We  have  therefore,  in  the  use  of  pencil  measurement  on  the 
picture  plane,  a  ready  and  accurate  means  of  ascertaining  any 
direction  or  any  proportionate  dimension  seen  by  the  eye.  It 
cannot  give  us  actual  sizes,  as  the  length  of  the  gable  in  feet ;  but 
it  will  tell  us  how  long  the  slanting  line  representing  the  gable 
must  be  drawn  in  proportion  to  other  parts  of  the  house.  In  this 
case,  for  instance,  the  sloping  edge  appears  three-fourths  of  the 
gable  width.  The  difficulty  in  using  this  valuable  aid  with  exact- 
ness lies  in  the  beginner's  trouble  in  keeping  the  pencil  always  in 
his  invisible  picture  plane.  Any  distance  between  the  eye  and 
the  object  may  be  assumed  for  the  picture  plane.  But  for 
accuracy  this  assumed  distance  must  be  kept  the  same  while 
comparing  sizes.  This  is  easily  done  by  sitting  erect  and 
measuring  at  arm's  length,  supporting  the  elbow  with  the  other 
hand  if  needful.  The  student  should  then  mentally  see  the  pic- 
ture plane,  recalling  that  it  is  vertical^  or  parallel  with  the  face 
when  looking  at  the  middle  of  the  objects  to  be  drawn.  That  is, 
it  is  at  right  angles  to  what  we  may  call  the  Central  Direction 
of  Seeing. 

The  Central  Direction  of  Seeing.  —  This  extends  from  the  eye  to 
the  center  of  the  objects  observed,  while  the  face  and  the  picture 
plane  are  parallel  to  each  other  and  at  right  angles  to  it.  The 
picture  plane  may  then  be  thought  of  as  a  transparent  vertical 
plane  pierced  in  its  middle  by  the  direction  of  seeing. 

We  have  said  the  central  direction  of  seeing  is  at  right  angles 
to  the  face.  Since  the  face  is  generally  vertical,  the  direction  of 
seeing  is  generally  horizontal  (A  in  Fig.  14,  Ch.  III).  The  com- 
monest exception  is  that  of  being  directed  slightly  downward  (B 
in  same  Fig.).  In  this  case  it  cannot  be  at  right  angles  to  the 
picture  plane.     It  will,  however,  always  appear  at  right  angles  to 

6 


PENCIL    MEASUREMENT,    ETC. 


it  when  looked  at  from  above.     That  is,  it  is  at  right  angles  from  side 

to  side,  and  in  a  plan  will  always  be  shown  at  right  angles,  as  in 

Fig.  8. 

Return  now  to  the  seat  (Fig.  7),  and  try  pencil  measurement 

on  the  turned  book.     Imagine  as  clearly  as  possible  the  trans- 
parent picture  plane  at  arm's  length, 

on  which  the  pencil  may  be  revolved, 

but  through  which  it  must  never  be 

thrust.     Starting  with  the  pencil  erect 

(Fig.  10)  drop  it  directly  over  to  the 

left    (Fig.   11),  watching   carefully  to 

keep  it  from  leaning  back  or  forward. 

Let  another  person  help  by  turning  the 

book  away  while  you  measure  it  and  at 

the  same  time  Fig.  lo 

keep  the  pencil  from  following  it  back- 
ward as  it  is  turned  away  (Fig.  12). 
Thus  as  the  book  is  turned,  the  pencil, 
if  it  remains  on  the  picture  plane,  shows 
^  the  book  to  appear  narrower  or  be  fore- 
shortened. What 


is  now  sought  for 
is  that  which  the 
eye  really  sees  as  the  width,  not  what  the 
mind  knows  it  to  be.  It  is  of  great  im- 
portance to  distinguish  sharply  between 
actual  facts  of  form  and  size  and  the  per. 
spective  appearance  of  them  as  presented 
to  the  eye.  Fig.  iz 

An  excellent  object  for  practice  is  a  door.  Stand  facing  a 
closed  door,  and  take  its  proportions  by  pencil  measurement. 
Then  let  some  one  open  it,  and  observe  the  apparent  decrease 
in  width. 

For  further  consideration  of  the  picture  plane  see  Chapters 
XXXIV,  XLI,  and  XLIII. 

7 


Fig.  11 


H 


Chapter  III 
THE    ELLIPSE 

AVING  learned  that  the  book  cover  and  door  appear 
foreshortened  in  proportion  as  they  are  near  to  co- 
inciding with  the   direction   in   which  they   are   seen, 


Fig.  12a 


we  naturally  look  for  the  same  change  in  the  Circle. 

Making  a  Cylinder. — Fold  

the  long  edges  of  a  piece  ^^ 
of  stiff  paper  (A  in  Fig. 
12a)  and  roll  it  into  a  cylin- 
der, tucking  one  end  of  the 
paper  under  the  folds  of 
the  other  (B  in  Fig.  12a). 

Seeing  the  Ellipse. — Holding  the  cylinder  vertically, 
as  in  A,  Fig.  13,  and  with  one  eye  closed,  raise  it 
slowly  till  on  a  level  with  the  eye.  The  top  now 
appears  as  a  straight  line  (B,  Fig.  13).  It  is  so  fore- 
shortened that  its  surface  is  entirely  lost  to  sights 
leaving  only  its  edge  visible.  Now,  keeping  the 
cylinder  vertical,  lower  it  till  the  eye  sees  into  it 
perhaps  half  an  inch.  Observe  carefully  the  shape 
formed 
by  the 
top. 
Turn  it 
so  the 
top  appears  as  a  cir- 
cle (A  in  Fig.  14), 
then,  iiolding  it  ver- 
tically again  (as  at 
B),  compare  men- 
tally the  apparent  shapes  as  the  top  is  placed  in  the  two  different 
positions.     Now  (keeping  it  always  vertical)  raise  and  lower  the 

8 


Fig.  14 


THE    ELLIPSE 


Fig.  15 


cylinder  slowly,  and   note   how  the   form  of  the  top  changes, 
appearing  rounder  as  it  is  lowered. 

Symmetry  of  the  Ellipse.  —  This  peculiar  shape,  varying  in  round- 
ness between  the  straight  line  and  the  circle,  represents  the  ap- 
pearance of  the  circle  seen  ohliqueli/,  and  is  the  Ellipse,  one  of  the 
most  beautiful,  spirited,  and  subtle  of  curves.  While  the  circle 
is  formed  by  a  curve  bending  equally 
in  all  parts,  the  outline  of  the  ellipse 
is  constantly  changing  in  the  degree  b( 
of  its  curvature.  From  the  middle 
of  each  side  (A,  A  in  Fig.  15)  this 
curvature  increases  smoothly  to  the 
ends  (B,  B).  Thus  the  ellipse  may  be  divided  by  lines  through 
the  middle  of  its  sides  and  ends  into  four  duplicate  curves  or 
quarters.  These  lines  are  known  as  the  Long  and  Short  Diiime- 
ters.  On  these  two  lines  the  ellipse  must  be  symmetrical,  what- 
ever the  proportion  of  the  diameters  to  each  other;  that  is, 
whatever  the  roundness  of  the  ellipse. 

Testing  the  Ellipse.  —  A  test  useful  to  determine  the  correctness 
of  a  drawing  of  the  ellipse  is  sighting  with  one  eye  along  the  long 
diameter.  If  the  ellipse  is  perfect  it  will  appear  foreshortened  to 
a  circle  having  a  diameter  equal  to  the  short  diameter  of  the  ellipse. 
But  there  is  no  test  of  the  ellipse  like  the  ellipse  itself  as  seen  in 
objects.  The  student  should  compare  his  drawing  of  ellipses  with 
the  rhythmically  varying  curves  which  compose  ellipses  as  seen  in 
real  objects,  correcting  and  comparing  till  the  eye  is  satisfied.  If 
this  be  faithfully  done,  the  time  will  be  short  before 
ellipses,  often  deemed  a  bugbear  of  freehand  draw- 
ing, become  a  pleasure  instead  of  a  penance. 

Roundness  of  Ellipses  According  to  Position.  —  Since 
the  top  ellipse  appears  rounder  as  it  is  dropped  below 
the  eye  level,  it  must  be  concluded  that  could  the 
bottom  be  fully  seen  it  would  appear  as  a  rounder 
ellipse  than  that  of  the  top.  Place  the  cylinder  on  the  table  and 
trace  around  the  bottom  with  a  pencil.     Move  the  cylinder  to  one 

9 


Fig.  16 


FREEHAND    PERSPECTIVE 


side  and  compare  the  shape  of  this  traced  ellipse  with  that  of  the 
top  ellipse  (Fig.  16).  Also  compare  both  with  that  part  of  the 
cylinder  bottom  which  can  be  seen.  There  is  no  difficulty  in 
perceiving  that  the  ellipses  in  a  vertical  cylinder  below  the  eye 
are  rounder  as  they  are  farther  below  the  eye  level. 

Now,  keeping  the  cylinder  vertical,  raise  it  slowly.  When  the 
bottom  ellipse  reaches  the  level  of  the  eye,  it  appears  as  a  straight 
line  (A  in  Fig.  17),  like  the  top  ellipse  when  at  the 
same  height.  When  the  cylinder  is  moved  on 
above  the  eye,  the  bottom  becomes  an  ellipse  (B), 
which  as  we  raise  it  farther  above  the  eye  level 
appears  rounder.  We  perceive  that  it  appears 
rounder  or  less  foreshortened  in  proportion  as  it  is 
farther  from  coinciding  with  the  direction  in  which 
the  eye  looks  to  see  it,  as  was  the  case  with  the 
book  cover  in  Chapter  II.  Furthermore,  if  the 
cylinder  be  turned  horizontally  and  held  at  the  level 
of  the  eye  with  its  length  parallel  to  the  picture 
plane,  and  one  end  be  brought  in  front  of  the  eye, 
we  shall  again  see  this  circular  end  as  a  straight  line  (B  in  Fig.  18), 
because  it  coincides  with  the  direction  of  seeing.  If  the  cylinder 
be  moved  horizontally  to  one  side,  still  keeping  its  length  parallel 
with  the  picture  plane  (A  in  Fig.  18), 
the  ellipse  appears  to  widen  exactly 
as  when  the  cylinder  was  held  verti- 
cally and  moved  above  or  below  the 
eye  level.  The  circular  top  appears 
as  a  circle  only  when  its  surface  is 
at  right  angles  to  the  direction  of 
seeing  (A,  Fig.  14).  When  oblique  to  this  direction,  as  at  B,  it 
appears  as  an  ellipse,  or  foreshortened  circle.  The  ellipse  is 
plainly,  therefore,  an  illustration  of  the  second  great  principle, 
that  of  Foreshortening. 

Practice  of  Ellipses.  —  The  student  should  now  practice  drawing 
ellipses,  both  vertical  and  horizontal,  until  they  can  be  formed 

10 


Fig.  18 


THE    ELLIPSE 

with  ease  and  exactness.  Mark  the  extreme  points  (A,  A,  B,  B 
Fig.  15)  first  taking  care  to  have  B  B  exactly  opposite  the  middle 
of  A  A.  Hold  the  pencil  for  drawing  ellipses  as  directed  m 
Chapter  I  for  straight  lines,  using  a  position  of  the  hand  that 
will  brmg  the  pencil  at  right  angles  to  the  long  diameter.  If  the 
elhpse  is  horizontal,  begin  it  a  little  to  the  left  of  the  middle  of 
the  upper  side,  drawing  to  the  right  first.  If  vertical,  begin  below 
the  middle  of  the  left  side,  and  draw  up.  Make  the  whole  outline 
with  one  movement,  first  carrying  the  pencil  evenly  several  times 
over  the  paper  without  touching  it,  to  gain  confidence  and  cer- 
tainty of  movement. 


11 


Chapter  IV 
A  CYLINDER   AND  A  CYLINDRICAL  OBJECT 

THE  student  should  draw  this  exercise,  following  carefully 
the  directions  given.     After  doing  so  he  should  draw  a 
cylindrical  object  of  his  own  choosing,  putting  in  practice 
the  principles  taught  in  this  chapter. 


KVSSMSafi^-^VAC*. 


Fig.  19 

Planning  the  Drawing.  —  The  Design,  or  Composition,  or  Decora- 
tive Arrangement  of  the  exercise,  which  is  that  kind  of  beauty 
secured  by  a  harmonious  and  artistic  relating  of  the  work  and  its 
spaces,  is  to  be  considered  first  in  all  drawings,  and  should  always 
be  kept  in  mind.  For  this  exercise  (Fig.  19)  we  first  consider 
how  to  place  most  effectively  in  a  drawing  these  two  separated 

12 


CYLINDER    AND    CYLINDRICAL    OBJECT 

objects,  a  cylinder  whose  height  is  twice  its  width  and  some 
simple  cylindrical  object  (in  this  case  a  rose  jar).  To  this  end 
after  drawing  the  margin  the  extreme  points  in  the  boundaries 
of  the  objects  are  lightly  indicated  on  the  paper  (Fig.  20),  taking 
care  that  the  spaces  between  them  and  the  margin  are  such  as  to 

give  an  agreeable  and  interesting  division  of  the 

inclosed  surface.     The  continuous  table  line  be- 
j  hind   them   indicates   that  they  stand   on  the 

same   surface,   and  thus   links   them  together. 

The  size  of  the  space  between  them,  being  no 

^°"  ^^  more  than  that   between  either   and   the   side 

margin,  also  helps  unite  them;   and  the  position  of  the  ornament 

on  the  jar,  near  the  middle  of  the  sheet,  attracts  the  eye  to  the 

center  in  comparison  with  the  whole. 

Drawing  the  Cylinder.  —  For  this  the  paper  cylinder  model  used 
in  Chapter  III  is  placed  as  shown  in  the  illustration  (Fig.  19), 
that  is,  a  little  below  the  eye  level,  and  at  least  six  times  its 
height  from  the  eye.  The  apparent  proportion  of  the  top  (that 
is,  if  the  width  of  the  ellipse  appears  to  be  one  third,  one  fourth 
or  some  other  part  of  its  length)  should  be  carefully  judged  by 
the  eye  and  then  tested  by  pencil  measurement.  Four  points  for 
this  ellipse  should  then  be  lightly  marked,  and  it  should  be  drawn 
through  these  points  as  previously  directed.  The  bottom  ellipse 
is  sketched  directly  under  the  upper  and  in  the  same  way,  re- 
membering that  it  must  be  of  the  same  length,  but  rounder.  The 
back  or  invisible  part  of  each  ellipse  is  left  light.  The  straight 
lines  for  the  sides  must  be  tangential  to  the  ends  of  the  ellipses, 
so  they  will  join  with  perfect  smoothness.  If  they  do  not  thus 
join,  the  ellipse  is  the  part  most  likely  to  be  wrong. 

Now  is  the  time  to  put  the  drawing  back  by  the  paper  model 
and  compare  the  two.  Look  longest  at  the  model,  glancing  briefly 
at  the  drawing ;  the  aim  being  always  to  form  in  the  mind  a  clear 
image  of  the  model's  true  shape,  and  to  correct  the  work  by  it. 
The  student  should  ask  himself  if  the  cylinder  in  his  drawing 
appears  to  press  evenly  on  the  ground  like  the  model.     The  com- 

13 


FREEHAND    PERSPECTIVE 

monest  error  is  that  of  bending  the  outline  of  the  partially  visible 
ellipse  too  much  at  A  in  Fig.  21,  and  not  increasing  the  curvature 
toward  its  ends  (B,  B),  thus  making  the  curve 
more  circular  than  elliptical,  and  causing  the  cyl- 
inder in  his  drawing  to  look  as  if  it  would  rock 
on  its  base,  instead  of  resting  firmly  on  every 
part  of  it.     Sketching  the  ellipse  entire  (C,  C)  i* 

Corr^c//<ys     ^^     ^^^     ^^     SVLGII     Si     CaSC. 

^^^'  ^^  Next  a  pencil  line  is  drawn  around  the  paper 

cylinder  half  way  between  the  top  and  bottom.  Points  (E,  E, 
Fig.  21)  are  then  marked  on  the  drawing,  one  half  way  between 
the  fronts,  and  the  other  half  way  between  the  backs  of  the 
ellipses.  These  points  should  be  tested  after  marking  and  made 
correct,  but  never  measured  till  the  eye  has  been  made  to  do  its 
utmost.  Observe  that  these  marks  give  a  short  diameter  for  the 
middle  ellipse  half  way  in  size  between  those  of  the  upper  and 
lower  ellipses. 

In  the  same  way  the  two  other  lines  around  the  cylinder  may 
be  made  on  the  model  and  represented  in  the  drawing. 

Now  another  paper  cylinder,  of  the  same  length  as  the  first 
but  only  two  thirds  its  diameter,  must  be  made,  and  placed  within 
the  first.  Have  the  space  between  them  even  all  the  way  around, 
so  that  the  circular  tops  of  the  two  cylinders  form  concentric  cir- 
cles. They  appear  as  ellipses.  Observing  carefully  the  space  be- 
tween these  ellipses,  the  student  easily  sees  that  it  appears  widest 
between  the  ends,  and  a  little  wider  between  the  front  sides  than 
between  the  back  sides.  As  we 
made  the  inner  cylinder  two  thirds 
of  the  diameter  of  the  outer,  the 
horizontal  space  between  the' ends 

Fig  ^^ 

of  the  two  ellipses  will  each  be  act- 
ually made  one  sixth  of  the  length  of  the  outer  ellipse.    They  will 
also  appear  as  sixths,  because  the  ends  of  the  ellipse  are  equally 
distant  from  the  eye.    The  ends  of  the  inner  ellipse   (C,  C)  are 
marked  by  light  vertical  lines.     For  its  front  and  back  we  divide 

14 


CYLINDER    AND    CYLINDRICAL    OBJECT 

the  width  of  the  outer  one  also  into  sixths,  but  as  these  sixths 
are  in  perspective  or  at  varying  distances  from  the  eye,  they  are 
perspective  sixths.  That  is,  they  appear  successively  smaller  as  they 
recede  from  the  eye.  This  perspective  division  is  here  made  wholly 
by  the  eye  (though  later  another  method  is  given).  The  per- 
spective middle  (point  G)  is  first  marked  on  the  short  diameter, 
making  the  near  half  considerably  larger  than  the  far  one.  Each 
perspective  half  is  then  divided  into  perspective  thirds,  after 
^  which  the  six  divisions  are  tested  to  see  if  they  are  successively 
smaller,  as  directed  above.  Draw  the  inner  ellipse,  making  its 
ends  tangential  to  the  vertical  lines  (C,  C),  and  exactly  opposite 
the  middle  of  its  short  diameter.  It  will  now  be  found,  if  we 
draw  the  long  diameter  of  each  of  these  two  ellipses  where  it 
must  always  be,  in  the  apparent  middle  from  front 
to  hack,  that  the  long  diameter  of  the  inner  one 
falls  higher  on  the  paper  than  that  of  the  outer 
one.  From  this  we  conclude  that  neither  long 
diameter  represents  the  actual  diameter  of  the 
circle.  Fig.  23  shows  the  plan  of  a  circle  with 
its  true  diameter,  A  A.  The  eye  at  x  sees  B  B,  a 
line  connecting  two  tangentials,  as  the  longest  line 
in  that  circle.  It  therefore  becomes  the  long  diam- 
eter of  the  ellipse  which  the  eye  in  that  position 
sees.  Meanwhile  the  actual  diameter  appears  both 
shorter  and  farther  back  than  B  B,  because  far- 
ther away.  That  part  of  the  circumference  back  ^^^-  -^ 
of  B  B,  though  actually  larger,  is  so  foreshortened  as  to  appear 
exactly  like  the  part  in  front,  producing  the  symmetry  which  is 
the  wonderful  and  unfailing  characteristic  of  the  ellipse. 

Since  the  inner  circle  is  smaller,  the  eye  can  see  farther  round 
it,  as  shown  in  Fig.  23.  This  furnishes  another  reason  for  its  long 
diameter  falling  farther  back,  and  agrees  with  the  fact  that  the 
really  even  space  between  the  two  circles  appears  greatest  in  front. 

The  Rose  Jar.  —  For  the  second  object  proceed  as  with  the 
cylinder,  drawing  lightly  all  of  the  ellipses  entire  first.     Should 

15 


FREEHAND    PERSPECTIVE 


Fig.  2J. 


any  fall  at  the  same  height  as  one  on  the  cylmder,  it  must  be 
made  of  the  same  roundness,  since  the  two  objects  are  shown 
by  the  table  line  to  be  on  the  same  surface,  and  are  equally  near 

the  eye.  Compare  the  size  of  the  ellipses 
with  the  extreme  width  of  the  space  occu- 
pied on  the  paper  by  the  jar.  Compare 
also  the  lengths  of  the  top  and  bottom 
ellipses,  and  the  length  of  each  with  the 
extreme  width  of  the  jar.  Observe  that 
the  sides  of  its  short  cylindrical  neck  slope 
outward  slightly  toward  the  body  of  the  jar. 
The  Shoulders  and  Base  of  the  Jar. — Before 
drawing  the  side  outlines,  hold  the  jar  verti- 
cally at  arm's  length,  and  with  the  top  on  a 
level  with  the  eye.  Mark  the  point  (A  in 
Fig.  2-i)  where  the  body  and  neck  boundaries  meet.  Holding  this 
point,  lower  the  object  till  the  top  appears  as  a  fairly  round  ellipse. 
It  will  be  plain  that  we  now  see  a  portion  of  surface  beyond  the  end 
of  the  ellipse,  and  more  than  half  way  over  its  shoulder.  The 
boundary  line  which  marks  the  limit  of  our  seeing  has  moved  back 
on  the  shoulder,  so  that  it  passes  out  of 
sight  behind  the  neck.  A  little  experi- 
menting shows  that  the  surface  visible  be- 
yond the  end  of  the  ellipse  is  in  exact  yl 
proportion  to  the  roundness  of  the  ellipse. 
Now  place  a  sheet  of  paper  on  the 
table,  and  first  holding  the  jar  so  its  base 
is  on  a  level  with  the  eye  (Fig.  25),  mark 
the  extreme  point  of  the  base,  B.  Lower 
the  jar  slowly,  till  the  bottom  rests  on  the 
paper.  Mark  point  B  on  the  paper  and 
then  the  points  (C,  C)  where  the  side 
boundaries  now  appear  to  meet  the  bottom  ellipse.  Trace 
around  the  bottom  and  lift  aside  the  object.  It  will  be  found 
that  the  projecting  mass  of  the  jar,  being  nearer  the  eye  than 
16 


CYLINDER    AND    CYLINDRICAL    OBJECT 

the  ellipse,  had  hidden  from  sight  more  than  half  of  it.  It  is 
also  evident  that  on  the  lower  and  receding  part  of  the  jar  the 
boundary  line  advances,  so  that  we  see  less  than  half  of  the  sur- 
face, instead  of  more,  as  at  the  top.  If  we  now  trace  on  the  jar 
its  boundary  line  and  turn  it  around,  the  tracing  will  be  seen  to 
cross  the  object  obliquely  (Fig.  26).  With  these  facts 
in  mind,  complete  the  drawing  of  the  rose  jar. 

Tangential  Joinings. — All  meetings  of  boundary  lines 
with  ellipses  must  he  tangential.  That  is,  they  must 
touch  so  that  if  smoothly  continued  they  would  not 
cut  the  ellipse. 
Artistic  Rendering.  —  The  jar  should  be  first  drawn  with  thin 
light  lines,  corrected  to  accuracy,  and  afterward  rendered  with  sig- 
nificance. For  though  outlines  are  entirely  conventional,  never 
being  seen  in  nature,  yet  they  may  not  only  be  made  to  inarl^:  off 
beautiful  and  interesting  shapes,  but  by  their  character  to  sug- 
gest other  truths  and  qualities  for  the  enhancement  of  charm. 
Thus  in  the  rose  jar  the  front  edge  is  shown  to  be  nearer  than 
the  back  by  the  heavier  line,  and  the  rounded  thickness  of  the 
top  is  indicated  by  the  absence  of  nearly  all  of  the  inner  ellipse 
at  the  back  and  of  the  outer  one  at  the  front.  The  sides  are 
drawn  with  a  little  lighter  lines  at  the  top,  and  though  firm 
enough  to  clearly  present  the  shape  of  the  jar,  are  lighter  than 
the  front  part  of  the  top,  because  representing  a  part  of  the  jar 
further  from  the  eye. 

The  ornament  may  help  to  express  the  rounded  form  of  the 
jar  by  its  foreshortened  shape  as  it  nears  the  boundary,  and  by 
the  greater  clearness  and  emphasis  of  that  portion  of  it  most 
thrust  forward.  Its  outlines,  emphasized  on  one  side,  with  the 
other  side  light  or  lost,  and  the  detail  shown  in  the  side  lines  of 
the  jar,  indicate  it  to  be  in  relief.  The  expression  of  color  in 
places  is  used  to  strengthen  the  projection  of  the  jar. 

These  remarks,  however,  must  not  be  understood  as  rules. 
They  are  but  suggestions  for  the  incitement  of  the  student  to 

use  his  own  artistic  judgment. 
2  17 


Chapter  V 


AN  OBJECT   ABOVE    THE   EYE   AND   THE 
CONE   PRINCIPLE 


T 


HE  electric  lamp  shade  here  shown  (Fig.  27)  is  vertical 
and  above  the  eye,  and  its  ellipses  therefore  increase  in 
roundness  toward  its  top.     The  student  should  draw  this 

exercise,  and  make 
another  study  from 
some,  object  similarly 
placed. 

Dra-wing  the  Object. 
I  —  Proceed  as  in  the 
\  previous    exercise. 
\  Observe  that  the  slop- 
\  ing   side   boundary 
I  lines  of  the  shade  join 
I  the  ellipses  in  front 
\  of   their   ends.      The 
\  flaring  shade  is   like 
I  the  lower  half  of  the 
rose  jar  reversed.    Its 
smaller    part    is    far- 
thest from  the  level 
of  the  eye,  as  was  the 
base  of  the  rose  jar, 
and  we  therefore  do 
not    see    half    way 
round   it.      To    make 
this  clear,  hold  a  cone^ 
in    various    positions 


Fig.  27 

^  One  may  be  made  from  paper  (Fig.  28). 
18 


OBJECT    ABOVE    THE    EYE,    ETC. 


Fig.  28 


(that  is,  above  and  below  the  eye),  and  with  apex  up  and  down 
(Fig.  29). 

The  button  through  which  the  cord  is  drawn  forms  an 
oblique  ellipse.  But  by  turning  Fig.  27  so  as  to  bring  this 
ellipse  horizontal,  the  button  will  be  found  symmetrical 
on  its  axis  (see  Ch.  XXVII);  and  as  it  is  arched  or 
thickened  in  its  middle,  the  space  in  front  of  the  holes 
for  the  cord  appears  a  good  deal  wider 
than  that  back  of  them.  Each  side  of 
the  button  is  a  very  flat  modified  cone. 
Notice  in  the  outline  of  the  cylinder  the 
slight  depression  marking  where  the  key 
enters  its  side. 

It  may  be  now  noted  that  ivhen  the  cone  apex 
(that  is,  its  decrease  of  diameter)  is  nearer-  the 
eye  than  its  base,  we  see  more  than  half  way  round 
it.  Conversely,  we  see  less  than  half  ivay  round 
tvhen  the  apex  is  farther  from  the  eye^.  This  char- 
acteristic of  curved  or  sloping  surfaces  in  cylindrical  objects 
may  be  termed  the  Cone  Principle.  Broadly  speaking,  it  is  the 
expression^  hy  outline  merely,  of  Belief,  or  Solidity,  or  TJiird 
Dimension.  The  rose  jar  in  Chapter  IV,  the  cream  jug^in  Chapter 
VI,  and  indeed  all  cylindrical  objects  with  flaring  or  bulging 
sides,  are  examples.  More  advanced  applications  of  this  principle 
are  found  in  the  drawing  of  such  natural  objects  as  trees  and 
mountains,  also  in  drawing  the  human  face  and  figure. 


Fig.  29 


19 


T 


Chapter  VI 
A  CREAM  JUG 

HE  Model.  —  Provide  an  object  similar  to  the  cream  pitcher 


here  shown  (Fig 
should  be  made. 


30),  from  which  the  student's  drawing 
If  inexperienced  he  will  be  helped  by 

„____ first   making   a   copy 

,  from  this  example. 
I  The  Handle.  — Place 
the  jug  a  little  below 
the  eye  (according  to 
the  directions  in  Ch. 
I).  Draw  the  cylindri- 
cal body  entire  first  as 
if  it  had  neither  handle 
nor  spout,  and  with 
light  lines  (Fig.  31). 
Then  hold  the  model 
with  its  center  at  the 
level  of  the  eye  and 
with  the  handle  in 
profile  (A  in  Fig.  32). 
Observe  that  a  center 
line  for  the  joining  of 
the  handle  with,  the 
model  would  fall  in 
the  curved  boundary 
line  or  profile  of  the 
jug.  Turn  it  to  bring 
the  handle  directly  in 
front,  when  this  same  center  line  {x  in  Fig.  32)  will  appear 
straight  and  vertical.     Now,  turn  the  jug  slowly  back,  bringing 

20 


t — 


Fig.  30 


A    CREAM    JUG 


Fig.  31 


Fig.  32 


the  handle  again  into  the  boundary  line.  It  is  apparent  that  as 
the  handle  revolves,  its  center  line  of  joining  changes  in  appear- 
ance from  a  straight  line  in  front  through  a  succession 
of  curves  that  increase  in  roundness  till  at  last  it  coin- 
cides again  with  the  profile  of  the  jug  (C  in  Fig.  32). 
These  curves  are  lines  such  as  would  be  produced  on 
the  surface  by  cutting  vertically  through  the  jug  cen- 

^  ter,  as  an  apple  is 

halved ;   and  may  be  named 
Profile  Lines  or  Profiles. 

Replace  the  model  on  the 
table  and  revolve  the  handle 
to  the  side  again,  when  it 
will  be  seen  that  the  ends 
of  these  profile  curves  rest  on  the  top  and  bottom  ellipses 
of  the  jug  (Fig.  33).  And  the  side  boundary  of 
the  jug  does  not  now  coincide  with  the  profile 
at  the  side,  as  it  did  (A  in  Fig.  32)  when  the  jug 
was  held  at  the  eye  level.  This  is  because  of 
the  change  in  the  position  of  the  boundary. 
As  the  jug  is  placed  below  the  eye  the  bound- 
ary advances  from  A  to  B,  recedes  from  B  to 
C,  and  advances  again  from  C  to  D,  in  accord- 
ance with  the  cone  principle  (Ch.  V). 
In  Fig.  34  is  shown  by  a  dotted  tracing  of  this 
boundary  how  it  actually  differs  from 
the  profile  curves  in  Fig.  33.  In 
sketching  a  profile  curve,  therefore,  allowance  must  be 
made,  as  shown  in  Fig.  33.  Note  how  x,  x,  x^  the 
points  of  the  smallest  diameter,  fall  in  an  ellipse  at  that 
height ;  also  the  points  ?/,  y,  y,  of  the  greatest  diameter. 
Fig.  34  q^^^  ^j  ij^^g^  profiU  curvcs,  shapcd  according  to  its 
nearness  to  the  boundary  of  the  object,  should  he  sketched  as  a 
guide  for  the  attachment  of  the  handle. 

In  the  same  way  we  observe  the  shape  of  the  handle  itself  to  vary 

21 


Fig.  33 


FREEHAND    PEKSPECTIVE 


Fig.  35 


Fig.  36 


according  to  position,  from  a  profile  vieiv  at  the  side  (A  in  Fig.  32) 
to  a  view  of  its  outer  surface  (B  in  Fig.  32). 

The  Spout.  —  Looking  directly  into  the  jug  from  above  (Fig. 
35),  we  note  that  the  spout  is  directly  opposite  the  handle,  so  that 
a  horizontal  line  through  the  middle  of  both  would 
pass  through  the  center  of  the  circular  top.  We  there- 
fore mark  the  perspective  middle  of  the  top  ellipse  (0 
in  Fig.  36)  (that  is,  making  the  nearest  half  larger), 
draw  a  line  through  it  from  the  center  of  the  handle  top, 
and  mark  the  end  of  the  spout  on  this  line.  For  the 
width  of  the  spout,  set  off  perspective  distances  from  this  line 
either  way  on  the  top  edge  of  the  pitcher,  remembering  that  the 
half  nearest  the  end  of  the  ellipse  is  much 
the  more  foreshortened;  and  that  the  dif- 
ference is  greater  the  more  the  top  is  fore- 
shortened. From  these  points  to  the  tip 
of  the  spout  straight  lines  may  be  sketched 
as  guides  for  drawing   the   edges,   which 

may  be  straight  but  usually  curve  both  upward 
and  sidewise.  The  profile  or  center  line  for  the 
spout  is  sketched  like  that  for  the  handle  (A  in 
Fig.  36). 

The  Foot.  —  The  drawing  of  the  foot  also  needs 
some  explanation,  though  covered  by  the  cone 
principle  of  Chapter  V.  In  profile  it  would  ap- 
pear as  at  A  in  Fig.  37,  with  the  circles  as  straight 
lines;  and  the  student  should  raise  his  model  to 
the  eye  level  and  observe  it  thus.  On  lowering 
the  model  these  seeming  straight  lines  appear  as  ellipses  (B, 
Fig.  37),  and  the  lower  part  of  the  side  boundary  lines  of  both  the 
pitcher  and  its  foot  move  forward  of  the  ends  of  these  ellipses 
till  tangential  joinings  are  made  at  C  and  D.  The  upper  part 
of  the  boundary  of  the  foot  moves  back,  joining  the  upper  ellipse 
at  E.  In  consequence,  this  side  outline  of  .the  foot  (E  D)is  a 
little  lengthened,  making  its  curve  less  round  than  in  profile. 

22 


Fig.  37 


A    CREAM    JUG 


the  other  with  the 
A 


The  lower  half  of  the  jug,  as  we  can  now  see,  is  a  modified  ver- 
tical cone  with  its  apex  down  (A,  Fig.  38).  The  foot  is  modified 
from  two  cones;  one  with  the  apex  up 
apex  down  (B,  Fig.  38).  See  Chapters 
ly  and  V. 

The  Ornament.  —  The  principle  fol- 
lowed in  suggesting  the  perspective  of 
the  ornament  will  be  readily  seen  from 

the  illustration  (Fig.  39).   The 

curved  guide  lines  are  parts 


ti\f  !<•  !  of  profiles  similar  to  those  for 
h'^^i  placing  the  handle  and  spout. 


Fig.  38 


The  student  will  now  begin  to  understand  that  it  is 
possible  to  recognize  and  suggest  the  solid  rounding 
surface  of  the  object  by  every  line  and  touch  upon,  it. 
To  this  end  that  part  of  the  ornament  nearest  the 
eye  is  more  emphasized  in  the  final  drawing.  And 
looking  carefully  at  the  object,  we  see  that  besides 
its  foreshortening,  that  part  of  the  ornament  near 
the  boundary  is  less  distinct,  and  is  often  lost  in  the  reflec- 
tions from  its  surroundings. 


\^'f 


K 


m 


Fig.  39 


23 


Chapter   VII 
A    TIME    STUDY 

HAVING-  carefully  studied  the  principles  of  cylindrical 
objects,  it  is  now  best  to  take  a  specified  time,  as  fifteen 
minutes,  for  the  more  free  drawing  of  such  an  object, 
choosing  a  simple  one  at  first.  Proceed  as  before,  except  that 
most  of  the  measuring  and  testing  must  be  omitted.     This  leaves 


Fig.  40.    A  Time  Study. 

time  to  draw  slowly  and  thoughtfully,  making  the  unaided  eye  do 
all  that  is  possible.  Study  the  general  shape,  looking  long  at  the 
object,  and  moving  the  pencil  several  times,  without  marking 
over  the  paper  where  the  lines  are  to  be  drawn  to  acquire  confi- 
dence and  certainty  of  touch.     Require  yourself  to  work  with 

24 


A    TIME    STUDY 

no  erasing  (except  of  such  construction  lines  as  may  show  when 
the  drawing  is  done),  and  to  stop  when  the  time  is  up.  It  will 
be  found  a  valuable  exercise  to  draw  in  this  way  the  same  object 
several  times.  After  one  drawing  is  done,  carefully  examine 
and  test  it  to  find  the  errors,  but  do  not  correct  them  on  that 
drawing.  Instead,  make  those  points  right  in  your  next  attempt 
at  the  same  object. 

Observe  in  Fig.  39  how  the  effect  of  glass  is  given  by  a  few 
lines  selected  from  those  many  graceful  curves  of  delicate  dark 
and  light  which  appear  in  the  object;  also  the  sketching  of  its 
high  lights,  or  window  reflections,  and  the  wavy  distortion  of 
lines  seen  through  it.  The  straight  lines  give  a  firmness  to  the 
composition  which  is  needed,  since  the  bowl  consists  wholly  of 
curved  lines. 


2.5 


Chapter  VIII 
A  GROUP  OF  CYLINDRICAL  OBJECTS 

FIG.  44  is  an  exercise  in  the  grouping  of  cylindrical  objects 
agreeably  and  appropriately  together.  The  student  is 
advised  to  first  draw  this  example,  using  carefully  the 
explanations  given.  After  that,  he  should  arrange  and  draw 
another  group  of  two  cylindrical  objects. 

Making  the  Composition.  —  In  composing  this   second   group, 
experiments  should  be  made  with  a  number  of  objects,  combin- 
ing them  in  different  ways.     A  Finder,  which  is  a 
card  having  a  small  rectangular  opening  cut  in  it 
(Fig.  41),  will  greatly  assist  in  judging  the  pictorial 


Fig.  41  effcct  of  a  compositiou,  especially  in  a  rectangular 
margin.  The  student  should  look  through  it  at  his  arrangement 
with  one  eye,  letting  its  edge  take  the  place  of  a  margin,  and 
moving  it  back  and  forth  till  the 
place  is  found  where  it  makes  the 
group  look  best.  Little  trial  or 
*' thumb-nail"  sketches  (Figs.  42 
and  43)  should  also  be  made  to 
determine  the  best  arrangement. 

In  Fig.  44,  for  example,  we 
observe  that  the  objects  are  such  as  might  naturally  be  placed 
together,  and  are  placed  in  positions  that  are  not  unusual.  Next 
their  shapes  make  a  pleasant  relief  or  contrast  to  each  other  with- 
out harsh  or  awkward  opposition ;  one  being  tall  and  slender  and 
the  other  lower  and  round.  Yet  the  teapot  is  not  so  low  nor 
wide  but  that  it  echoes  in  some  degree  the  dominant  height  of 
the  candlestick,  thus  aiding  harmony.      Its  spout  is  allowed  to 

26 


A    GROUP    OF    CYLINDRICAL    OBJECTS 

project  across  the  candlestick,  thus  contributing  to  the  unity  of 
the   composition.      The  leaning  bowl,  by  passing  behind  both, 
also    strengthens 
unity,  and  by  its 
lighter  and  more 
interrupted  lines 
furnishes  a  tran- 
sition, or  connec- 
tion, between  the 
nearer      objects 
and     the     white 
paper.       These 
results  might   be 
secured  by  other 
groupings.      But 
had    the    candle- 
stick   been    in 
front,    for    in- 
stance,   its     pro- 
jection above  and 
below  the  teapot 
would  have  been 
so  nearly  equal  as 
to  seem  uninter- 
esting   (Fig.   42). 
Yet  we  could 
have   remedied 
this  somewhat  by 
placing  the   can- 
dlestick a  greater 

distance     in    ad-  Fig.  44 

vance,  or  raising 

the  teapot  handle.  Or  the  cover  could  have  been  placed  on  the 
ground  in  front  (Fig.  43)  —  indeed,  many  possibilities  will  be 
suggested  by  a  little  study. 

27 


FREEHAND    PERSPECTIVE 

Drawing  the  Group.  —  In  drawing  this  exercise,  observe  that 
though  the  bottom  ellipses  of  the  two  objects  are  at  the  same 
level  the  nearer  appears  slightly  rounder  (Solutions  of  Problems, 
p.  162).  Care  must  be  taken  in  placing  these  ellipses  to  allow 
for  the  bulging  of  the  teapot  sides.  Remember  that  in  propor- 
tion as  the  bottoms  are  drawn  foreshortened  so  must  all  spaces 
on  the  table  be  regarded  as  foreshortened.  Note  also  that  all 
ellipses  in  the  candlestick  which  are  nearer  the  eye  level  than  the 
top  of  the  teapot  will  be  less  round  than  those  of  the  teapot. 

The  Teapot  Ears.  —  In  placing  these,  an  ellipse  may  be  used  as 

a  guide  (A  in  Fig.  45) .     The  middle  points  of  the  two  ears  should 

A  ^--:5^"~~-v,       ^®  ^^  ^  ^^^^^  passing  through  the  perspective  (that  is, 

^V^^^S^w  actual)  center  (o)  of  this  ellipse,  as  were  the  handle 

'y^^-- — ^^;\>1  and  nose   of  the  pitcher  in  Chapter  VI.      The 

J3   ^,j'^2_^^        cover  is  arched  (B  in  Fig.  45),  so  that  it  conceals 
J—  — i.     the  back  of  its  elliptical  edge.    This  arched  shape 

PROFILE   VIEW  .,..,  •  ^  o  o     • 

6How.NGARCHOFcovtR    jg  distuictly  socu  111  tiiG  form  of  its  top  boundary 

and  is  a  very  different  shape  from  the   ellipse. 

But  this  arched  boundary  does  not  fall  in  the  actual  middle  of 

the  cover  (since  we  are  looking  down  on  it),  but  a  little  beyond 

that.     The  knob  is  in  the  actual  middle. 

The  rendering  of  two  things  is  more  complicated  and  inter- 
esting than  that  of  one  alone.  As  the  candlestick  is  farther  away 
than  the  teapot,  its  lines  are  made  lighter,  and  in  places  are  quite 
lost.  The  lines  of  the  glass  rim,  or  hoheche  are  thinner,  more 
interrupted  and  more  smoothly  sweeping.  The  leaning  bowl 
may  be  omitted  at  this  time,  if  found  difficult.  The  principles 
of  its  construction  are  given  later  (Ch.  XXVII).  If  omitted  it 
will  be  found  necessary  to  make  the  farther  lines  of  the  candle- 
stick lighter  yet,  to  serve  in  place  of  the  bowl  as  a  transition. 


28 


Chapter  IX 

CYLINDRICAL  OBJECTS   GROUPED 
WITH  FRUIT 

A  N  example  of  grouping  is  here  given  in  which  part  of  the 
/\      group  is  cut  by  the  margin,  while  the  apples  illustrate 
-A-   ^   the  combination  of  natural   forms  with  cylindrical  ob- 
jects.   As  in  the  preceding  exercise,  the  student  should  compose 


Fig.  46 

a  group  corresponding  to  this  exercise  and  draw  it ;  and  if  inex- 
perienced should  draw  this  before  making  his  original  one. 

Study  of  the  Group.  —  In  locating  the  pitcher  on  the  paper,  see 
that  its  base  is  far  enough  from  the  dish  for  the  two  objects  to  clear 

29 


FREEHAND    PERSPECTIVE 

each  other.  Observe  the  generally  elliptical  shape  of  the  curves 
in  the  glass  pitcher ;  and  how  the  edge  of  the  plate  is  seen  warped 
and  interrupted  through  it.     The  plate  is  made  subordinate,  as 

forming  part  of  the  background  for 
the  other  two  objects.  Its  position, 
appearing  in  its  actual  shape  as  a 
simple  circle,  contributes  to  the  de- 
sired effect  of  quietness  or  subor- 
dination, as  does  its  being  .cut  by 
the  margin  line,  and  its  lighter  and 


\ 


\ 


PESWtCTIVE. 


X-Y      roRESHORTENED. 


Fig.  47  slightly  interrupted  lines. 

Since  it  is  standing  vertically,  it  must  be  supported  by  a 
vertical  surface  behind  it.  Consequently  the  table  line  (Ch.  I) 
must  be  placed  only  far  enough  on  the  paper  above  the  lowest 
point  of  the  plate  edge  to  express  the  foreshortened  necessary 
distance  of  this  point  from  the  wall  behind  it  (Fig.  47). 


30 


Chapter  X 


A  GROUP   OF   OBJECTS  FROM  MEMORY 
OR  INVENTION 

THIS  example  illustrates  the  drawing  of  objects  from 
invention  or  memory.  The  student  may  sketch  this 
exercise  as  directed;  then  should  invent  or  draw  from 
memory  one  of  his  own  arrangement,  making  small  trial 
sketches  as  in  Chapter  YIII,  and  using  the  best  of  these  in  his 


:-;;>:>^,---fC==CXAM\\ 


_ -..J 


Fig.  48 

final  composition.  Should  his  memory  not  be  clear  enough 
for  this,  it  may  be  refreshed  as  often  as  necessary  by  study 
of  the  objects  he  chooses  to  draw,  the  only  condition  being 
that  the  drawing  he  done  ivithout  the  object  in  view. 

31 


FREEHAND    PERSPECTIVE 


Drawing  the  Above  Study.  —  In  this  exercise  the  Japanese 
luncheon  carrier  is  placed  first.  Its  ellipses  are  sketched  in 
full,  whether  entirely  seen  or  not.  The  bowl-shaped  top,  being 
slightly  inclined,  is  drawn  on  a  leaning  axis  (A  B  in  Fig.  49). 
But  it  is  perfectly  symmetrical  on  this  axis  (Ch.  XXVII) .  This 
symmetry  should  be  tested  in  the  drawing  by  turning  it  to 
bring  the  axis  vertical,  when  any  error  is  easily  detected.  (Ch. 
XXVIII.) 

The  Flat  Dish.  —  It  was  desired  to  draw  the  flat  dish  as  it 
would  appear  if  touching  the  luncheon  carrier.  Its  height  (x  y) 
is  therefore  measured  upon  the  front  of  that  object  from  its  lower 

edge,  and  an  ellipse  of 
the  proper  roundness 
drawn  at  that  height. 
The  top  ellipse  of  the 
dish  would  touch  the 
other  object  somewhere 
in  this  ellipse,  and  so 
was  drawn  tangential 
?i  to  it.  To  obtain  the 
bottom  ellipse  of  the 
dish,  this  same  height, 
increased  to  allow  for 
its  slightly  greater  near- 
ness to  the  eye,  was  measured  downward  from  the  dish  top. 
But  as  the  sides  of  the  dish  are  flaring,  this  measuring  was  done 
from  the  estimated  true  middle  (0  in  Fig.  49)  of  the  top  of  the 
dish,  giving  O'  for  the  true  center  of  the  lower  ellipse.  The  foot 
is  like  a  very  short  cylinder.  The  flaring  sides  of  the  dish  are 
drawn  tangentially  from  the  rim  (F,  F)  to  the  upper  ellipse  of 
the  foot. 

The  Ornament. —  In  drawing  the  ornament  on  the  luncheon 
carrier  the  explanation  in  Chapter  VI  is  recalled.  On  the  cover 
the  band  of  fret  decoration  appears  narrowed  at  its  front,  and 
widest  at  the  ends.    It  is  a  modification  of  the  cylinder  top  in 

32 


Fig.  49 


OBJECTS    FROM    MEMORY 

Chapter  IV.  Note  the  foreshortening  in  its  details,  and  how  the 
Hnes  of  the  fret  express  the  curving  form  of  the  cover.  It  will 
be  seen  that  the  stripes  on  the  object  and  some  lines  of  the  fret 
follow  the  profile  lines  mentioned  in  Chapter  VI. 

The  Fan.  —  Like  the  plate  in  Chapter  IX,  the  fan  is  purposely 
placed  so  that  it  is  not  foreshortened.  Therefore  the  two  points 
(G,  Gr)  at  which  it  rests  on  the  table  appear,  as  they  actually  are, 
in  a  horizontal  line.  It  also  appears  in  its  true  shape,  symmetrical 
on  an  axis  passing  through  its  handle  (H,  H).  It  is  more  easily 
drawn  entire  first,  erasing  later  the  part  not  needed. 


"^ 


33 


Chapter  XI 

THE  CYLINDER  CONE  AND  BALL  GROUPED 
—A  PROBLEM  FOR  ORIGINAL  STUDY 

GENERAL    Conditions   for   Perspective   Problems.  —  Problems 
are  to  the  student  both  a  test  of  his  comprehension  of 
the  subject  thus  far,  and  an  exercise  by  which  the  subject 
becomes  firmly  fixed   in   his   mind.     To   this  end  the  drawings 
must  be  made  tvitliout  the  models  in  sight,  though  they  should  be 
studied,  and  if  necessary  even  sketched  in  the  required  positions 
before  drawing.     If  the  student  is  at  loss  to  recall 
their  appearance  while  engaged  in  work  they  may 
be  studied  as  often  as  needed ;  provided  only  that 
neither  the  models  nor  sketches  of  them  are  be- 
fore the  student  as  the  drawing  is  made.     It  can- 
'  ncTUBc'^  Pu^^4E  )'    not  be  expected  that  any  object  should  be  drawn 
'  '  '     until  opportunity  has  been  given  for  its  thorough 

^^^-  ^0  study ;  but  on  the  other  hand  it  is  not  mastered 

until  it  can  be  correctly  drawn  from  unaided  knowledge  and 
memory.  The  stated  dimensions  are  important,  giving  training 
in  the  expression  of  proportion,  though  drawings  need  not  be 
full  size. 

Drawings  should  of  course  be  made  without  assistance,  and 
without  referring  to  the  explanations  in  the  back  of  this 
book.  When  the  student  has  under  the  required  conditions 
made  his  drawing,  he  may  then  test  his  work  by  consulting 
the  explanation. 

Conditions  of  this  Problem.  —  In  this  problem  the  cylinder  and 
cone  are  to  be  4"  in  diameter  by  8"   high,  and   the   ball  4''  in 

34 


CYLINDER    CONE    AND    BALL 

diameter.  The  group  is  to  be  drawn  as  if  resting  on  a  surface 
which  is  twice  the  cylinder  height  below  the  eye,  and  at  least  six 
times  its  height  distant.  The  cylinder  stands  on  one  end  and  the 
cone  on  its  base,  touching  the  cylinder  and  a  little  in  front  of  it 
at  one  side.  The  ball  also  touches  the  cylinder,  and  is  a  little 
more  in  front  of  it  on  the  other  side.  The  plan  (Fig.  50)  will 
make  this  clearer. 


S5 


Chapter  XII 


THE    STUDY   OF   STRAIGHT-LINE 
OBJECTS 

A  Book  with  Back  Parallel  with  the  Face 

FOR  this  study  provide  a  book,  two  long  pencils,  and  three 
yards  of   fine   twine,  also   paper  for   sketching.     Choose 
a  book   of   interesting   appearance;     a   somewhat   worn, 
leather-bound  book  is  best.     Place  it  well  back  on  the  table  in 

front  of  you  and  below  the 
eye  with  its  back  next  to 
and  parallel  with  the  pic- 
ture plane,  and  its  ends 
equally  distant  from  you 
(Fig.  51). 

The  Book  Below  the   Eye. 

—  Two  surfaces  are  visible,, 
the  back  and  one  cover. 
Count  the  edges  seen  (seven), 
then  decide  how  many  of 
these  are  actually  horizontal.^ 
If  the 

Fig.  51 


book  is 
placed  as  directed,  its  back,  being  parallel  with 
the  picture  plane,  will  be  seen  in  its  true  shape 
if  traced  upon  it.  Lifting  the  cover  till  it 
is  vertical  (Fig.  52),  we  see  that  the  cover 
also  now  appears  in  its  actual  form 

^  It  may  not  at  first  be  realized  that  the  ends  of  the  cover  are  horizontal,  as  M'ell  as  its 
sides.  But  as  they  are  contained  in  a  horizontal  surface  (in  this  case  the  cover),  they  also 
must  be  horizontal.  Their  perspective  appearance  must  be  distinguished  from  their  actual 
position. 

36 


Fig.  52 

But  as  we  drop  it  slowly 


STRAIGHT-LINE    OBJECTS 


back  till  horizontal,  we  observe  that  the  further  edge  seems  to 
grow  shorter  because  moving  from  the  eye,  and  that  the  whole 
cover  becomes  foreshortened  or  narrowed  from  front  to  back, 

like  the  circular  ends  of  the  cylinder  in 
Chapter  IV.  If  (as  with  the  house  in 
Ch.  II) ,  a  pane  of  glass  were  standing  in  place 
of  the  imaginary  picture  plane,  a  tracing  of 
the  cover  on  that  would  be  a  true  perspec- 
How  to  draw  on  the  paper  such  a 
Stand  the  pen- 


FiG.  53 


V  tive  of  it. 
perspective  is  our  problem. 


Fig.  54 


cils  against  the  nearest  corners  of  the  cover 
(Fig.  53) ;  then  closing  one  eye,  and  keeping  the  other  exactly 
opposite  the  middle  of  the  book,  incline  the 
pencils  toward  each  other  (being  careful 
not  to  lean  them  back  or  forward)  until 
they  appear  to  lie  just  along  the  retreat- 
ing ends  of  the  cover  (Fig.  54).  Let 
another  person  hold  a  ruler  against  the 
pencils,  moving  it  down  until  its  edge  seems 
to   coincide  with  the  further  edge  of  the 

cover  (Fig.  55).  Now  the  pencils  and  the 
ruler  together  picture  the  apparent  shape  of 
the  cover,  and  we  plainly  see  how  the  ap- 
parent shortening  of  the  back  edge  (caused  by 
its  greater  distance  from  us)  makes  the  ends 
appear  to  converge  toivard  each  other.  The 
question  now  is :  Can  the  law  of  that  conver- 
gence be  so  determined  that  it  may  be  applied  in  any  drawing! 

The  Converging  Book  Ends. —  Substitute  for  the  pencils  the 
string  slipped  under  the  cover  to  the  back,  and  using  one  eye  as 
before,  bring  the  ends  together  so  that  the  strings  will  appear  to 
exactly  coincide  with  the  ends  of  the  cover  as  did  the  pencils. 
(Be  sure  to  keep  the  string  vertically  over  the  front  edge  of  the 
book,  not  letting  it  fall  back  or  forward.)     The  pencil  may  now 

37 


Fig.  55 


4^zm 


FREEHAND    PERSPECTIVE 


be  taken  in  the  other  hand,  and  slipped  down  on  the  string,  to 
form  again  the  shape  of  the  foreshortened  cover  (A  in  Fig.  59). 

Still  holding  the  string  as  before,  raise  the  book  and  string 
a  few  inches,  keeping  the  book  level  and  the  string  taut  (A  in 


Fig.  56 

Fig.  56).  The  string  does  not  now  cover  the  book  ends,  and 
the  joining  must  be  brought  lower  (as  in  B)  that  it  may  do  so. 
If  the  book  is  raised  more,  the  joining  is  yet 
nearer  to  the  book,  as  in  C;  until  when  the 
book  cover  is  at  the  level  of  the  eye  (Fig.  57) 
the  string  and  the  book  cover  both  disappear 
in,  or  coincide  with,  the  upper  edge  of  the  book. 
Now,  starting  with  the  position  last  shown, 
(Fig.  57)  hold  the  thumb  and  finger  firmly  at 
that  place  on  the  eye  level  (this  can  be  done 
by  noting  a  point  behind  it  on  the  wall),  and 
let  the  book  drop  slowly.  Keep  it  exactly 
horizontal,  and  let  the  string  slip  through  the 
stationary  thumb  and  finger,  so  that  their  meet- 
ing point  remains  at  the  eye  level.  If  this  is  carefully  done,  it 
will  be  seen  that  as  the  book  descends,  the  string  continues 
to  cover  the  converging  ends  (as  in  C  and  B,  Fig.  56).  At  the 
same  time  the  cover  appears  to  grow  wider,  and  its  ends  more 
and  more  nearly  vertical. 

38 


Fig.  57 


STRAIGHT-LINE    OBJECTS 


These  experiments  should  also  be  tried  with  the  lower 
cover,  holding  the  book  above  the  eye  (Fig.  58),  and  raising  and 
lowering  it. 

From  the  foregoing  study  it  is  easily  perceived  that,  provided 
we  Jceej)  the  book  horizontal,  the  point  toward  ivhich  its  ends  appear  to 
converge  remains  always  at  the  level  of  the  eye. 
We  have  therefore  only  to  sketch  the  eye  level 
at  its  right  height  compared  with  some  measure- 
ment on  the  object  and  mark  the  point  of  con- 
vergence in  the  right  place  on  it,  to  be  able  to 
use  it  for  drawing  these  converging  lines. 

We  have  also  found  that  the  horizontal  booh 
covers  (like  the  cylinder  top  in  Ch.  IV)  appear 
foreshortened  according  as  they  approacli  the  eye 
level,  whether  above  or  below  it  (Figs.  ^6 
and  58). 

Sketching  the  Book.  —  The  book  may  now  be  replaced  as  at  first. 
Then,  holding  the  strings,  as  before,  take  the  pencil  as  in  B,  Fig. 


Fig.  58 


59,  that  the  thumb  nail  may  be  used  as  a  sliding  gauge.  With  it 
measure  the  length  of  the  back  of  the  book  on  its  upper  near 
edge  and  compare  its  length  with  the  vertical  distance  from  this 
edge  to  where  the  strings  join  (C  in  Fig.  59).  (In  this  case  it 
takes  one  and  one  fourth  of  the  book  length  to  reach  the  joining 
of  the  strings.)  Now  the  back  of  the  book  may  be  sketched  in, 
the  point  of  convergence  (the  joining  of  the  string)  marked  on 

39 


FREEHAND    PERSPECTIVE 


Fig.  60 


the  paper  one  and  a  half  book  lengths  above  its  middle,  and  lines 

drawn  from  the  upper  corners  of  the  back  to  this  point.  On 
these  lines  the  ends  of  the  cover  are  to  be  marked 
off.  The  perspective  or  apparent  width  of  the 
cover  may  be  found  by  measuring  it  with  a  pen- 
cil held  vertically  as  in  Fig.  60,  and  comparing 
this  dimension  with  the  length  of  the  book.  In 
this  case  the  apparent  width  is  one  fourth  of  the 
book  length. 

The  Level  of  the  Eye.  —  This    will   be    found    of 
the  greatest  importance  in  all  drawings.     It  should 

be  carefully  marked  in  the  drawing  as  soon  as  the  position  of  the 

objects  on  the  paper  give  a  basis  for  locating  it.     At  first,  another 

person  may  assist  (Fig.  61),  but  a 

little  practice  will  enable  the  student 

to  find  it  for  himself.      The  top  of 

a   pencil,    held    vertically   over    the 

objects,    will    appear    as   a   straight 

line  when  at  the  height  of  the  eye 

(Fig.   62).      Or   if   any   part   of  the 

study  is  as  high  as  the  eye,  the  eye 

level  will  be  where  any  horizontal  Fig.  ei 

surface  or  any  receding  horizontal  lines  appear  as  straight  lines. 

See  Fig.  63. 

Parallel  Lines.  —  By  holding  one  string  down  on 

the  near  end  of  a  margin  line  on  the  book  this  line 
will  be  seen  to  converge  to  the  same  point  with 
the  two  ends  (Fig.  64).  By  placing  a  second  book 
on  and  parallel  to  the  first,  we  can  show  that  all 
lines  parallel  until  the  fir^st  two  converging  ones  ivill 
appear  to  converge  ivith  them  to  the  same  point.  An 
Fig.  62  important  deduction  from  this  is  that  parallel  lines 

appear  to  converge  to  the  same  point. 

It  is  also  evident  that  since  the  whole  book  cover  is  fore- 
shortened from  front  to  back,  the  margins  will  be  foreshortened 

40 


STRAIGHT-LINE    OBJECTS 


ill  the  same  direction.  And  we  find  that  the  side  margins  are> 
foreshortened  in  length,  but  not  in  width;  while  the  front  and' 
back  margins  are  foreshortened  in  width,  and 
the  back  one  more  than  the  front.  This  fol- 
lows the  principle  of  the  top  of  the  hollow 
cylinder  in  Chapter  IV. 

The    Vanishing    Point.  —  We    see    that    the 
book   ends   seem   to   converge  in   proportion  ^^^-  ^^ 

as  the  back  edge  of  the  cover  appears  shorter.      If   a  second 
book  like  this  were  placed  back  of,  and  touching  it,  its  front 

edge  would  appear  of 
the  same  length  as  the 
back  of  this,  and  its  back 
edge  shorter,  while  its 
ends  would  converge  in 
a  line  with  those  of  the 
first  book  (Fig.  65).  This 
can  be  imagined  as  re- 
peated infinitely,  each 
book  appearing  smaller 
^^^-  6^  than  the  one   before   it, 

and  the   cover  ends  all  falling   in   the  same  con- 
verging lines,  until  a  point  would  be  all  that  could 
rei3resent   the    last  .  book.    .The 
row  of  books  might   be  said   to 
vanish  in    this   point,   which   is 
therefore    called    the    Vanishing 
Point  of  such   lines  as   converge 
toward  it,    as    do    the    ends    of 
the     book     cover.     Vanishing 
Points,   like    the    Level   of    the 
Eye,  play  a  most  important  part  in  the  study 
of  perspective. 
A  familiar  example  of  vanishing  lines,  as  those  which  appear 
to  vanish,  or  converge  perspectively,  are  called,  is  found  in  a 

41 


FREEHAND    PERSPECTIVE 

receding  railroad  track  (Fig.  66).  The  ties  appear  shorter 
as  they  are  successively  farther  from  the  eye;  and  the  rails 
appear  and  converge,  till  the  whole  track,  if  it  could  be  seen 
for  a  long  enough  distance,  might  seem  to  disappear,  or  vanish 
in  a  point. 


42 


Chapter  XIII 
DRAWING  THE  BOOK  IN  TWO  POSITIONS 


T 


HE  student  may  copy  this  example,  but  in  any  case 
should  place  a  book  successively  in  these  positions  and 
draw  from  that;  having  it   high  enough  or  far  enough 


from  the  eye,  to  see  it 
in  a  normal  position  as 
explained  on  page  2. 
He  should  also  make 
drawings  from  mem- 
ory of  a  book  in  both 
positions,  expressing 
them  as  artistically 
as  possible. 

In  the  first  position 
on  this  sheet,  the 
eye  level  falls  off  the 
paper;  and  may  be 
marked  for  use  on  a 
piece  of  paper  fastened 
to  the  drawing  (Fig. 
68).  See  that  the  table 
line  is  high  enough  on 
the  paper  to  clear  the 
lower  back  corners  of 
the  book. 

It  will  be  observed 
that  the  back  of  the 
book  is  not  quite  flat 
but  slightly  curved  —  a  modification  of  the  cylinder, 
be  understood   by  holding   the   cylinder  horizontally 

43 


L 


Fig.  67 


This  will 
(Fig.  69). 


Fig.  68 


FREEHAND    PERSPECTIVE 

The  curve  opposite  the  eye  is  seen  as  a  straight   line,  since  it 
coincides  with  (or  lies  in  a  plane  passing  through)  the  direction 

of  seeing.  The  farther  these  lines  are  from 
coinciding  with  this  direction  (in  this  case 
to  right  and  left)  the  more  apparent  is  their 
curvature. 

For  the  second  position  in  this  exercise 
the  book  is  opened  and  turned  around  so  that 
its  ends  are  parallel  with  the  picture  plane. 
They  may  therefore  be  drawn  in  their  true 
shape  like  the  back  of  the  book.  The  sides 
and  all  lines  parallel  with  them  now  vanish 
to  VP  ^  (on  the  eye  level  directly  in  front  of 
the  student).  Note  that  points  A,  B,  and  C, 
where  the  book  rests  on  the  horizontal  table  (Fig.  68),  are  in  a 
straight  line  that  is  parallel  to  the  picture 
plane,  and  therefore  drawn  in  its  true 
direction,  which  is  horizontal.  Observe 
the  projection  of  the  covers  beyond  the 
leaves,  and  that  it  extends  backward  at 
D  and  E.  The  thickness  of  the  covers 
must  be  recognized,  though  the  wearing  off  of  the  edges  and 
corners  may  obliterate  their  sharpness.  Since  the  right  and  left 
corners  of  the  book  are  equally  distant  from  the  eye  care  must 
be  taken  that  the  covers  are  drawn  of  equal  width.  The  clasps 
must  be  long  enough  to  allow  of  their  being  fastened  when  the 
book  is  closed.     Their  ends  are  in  a  line  converging  to  VP. 

The  table  line,  being  a  subordinate  element,  should  be  so 
placed  that  most  of  its  length  is  covered.  Avoid  anything  which 
would  tend  to  emphasize  it,  as  making  it  coincide  with  the  back 
corners  of  the  book. 

1  Used  as  an  abbreviation  for  the  vanishing  point. 


Fig.  69 


44 


Chapter  XIV 
THE  BOOK  WITH   A   CYLINDRICAL  OBJECT 

THIS  exercise  (Fig.  70)  combines  a  book  in  one  of  the 
two  positions  previously  studied  with  a  cylindrical  ob- 
ject. The  student  may  draw  this  example  or  not, 
according  to  his  proficiency;  but  should  compose  and  sketch 
a  similar  group,  arranging  and  making  trial  sketches  of  several 


Fig.  70 

compositions.  Observe  that  the  extreme  points  of  the  book 
must  be  equidistant  from  the  eye,  as  in  Chapters  XII  and 
XIII.  But  as  soon  as  we  place  another  object  with  the  book, 
the  two  must  be  considered  together  as  forming  one  group  or 
picture. 

45 


FREEHAND    PERSPECTIVE 


PUN  OF  AJ 


9HOWINO 

PLANE 

TO  BOOK 


This  brings  us  to  reflect  that  whatever  the  number  of  objects 
we  include  in  our  picture,  it  is  always  drawn  tvith  the  eye  directly 
opposite  the  picture  as  a  whole,  so  that  the  center  of  seeing  is  in 
the  middle  of  the  group  from  side  to  side.  We  also 
recall  that  the  picture  plane  is  always  at  right 
angles  (viewed  from  above)  to  the  direction  of  see- 
ing. So,  if  the  cylindrical  object  is  placed  on,  or 
in  front  of  the  book  (as  in  Figs.  70  and  71),  the 
central  direction  of  seeing  the  picture  is  not 
changed ;  and  the  picture  plane  continues  parallel 
to  one  set  of  lines  in  the  book  as  in  the  preceding 
exercise  with  the  book  alone.  (See  plan,  Fig.  71.) 
If,  on  the  other  hand,  the  cylindrical  object  is 
placed  at  the  side  and  the  picture  thus  enlarged 
in  one  direction  only  (Fig.  72),  the  direction  of 
seeing  is  immediately  thereby 
moved  to  correspond,  and  the 
picture  plane  moves  with  it. 
The  book  will  cease  to  be  equi- 
distant at  its  ends  from  the 
picture  plane  and  cannot  be  drawn  as  previ- 
ously studied.  This  subject  is  considered 
more  fully  in  Chapters  XXXIV  and  XLI. 
It  would  of  course  be  possible  to  add  objects 
to  the  book  equall}^  at  both 
sides  (as  in  Fig.  73),  but  dan- 
ger of  stiffness  in  such  an 
arrangement  must  then  be 
remedied  by  some  such  device  as  the  string  of 
beads,  making  a  more  complicated  study  than 
is  desirable  at  present. 
For  this  exercise,  therefore,  place  the  cylindrical  object  some- 
where within  the  extreme  points  from  side  to  side  of  the 
book. 

It  will  be  observed  that  a  cylindrical  object  is  always  placed  so 

46 


Fig.  71 


PlANOrX 

OSJECTSX 
PLACED    \ 

M 
^ 

/the  SAKIE 
/with  the  bOOK 
/at  TKt  RiOHT. 

TIltOEMTIM 
13  MOVED,    \ 

J 

/of  the  PICTUOC 
/and  TMt  PICTvnC 

TDTTIB     ' 

/loNOSH    PARALLtl. 
/SOOK. 

\L 

Fig. 

72 

Fig.  73 


THE    BOOK    WITH    A    CYLINDRICAL    OBJECT 

that  a  part  of  its  base  is  seen,  if  only  a  very  small  part.  For 
this  reason,  it  is  not  put  behind  the  book  unless  the  foot 
can  be  left  partly  visible.  The  reason  for  this  precaution  is 
the  uncertain  effect  produced  by  a  study  in  which  it  is  not 
observed. 


Chapter  XV 

THE  CYLINDER  AND  RECTANGULAR  BLOCK 
-  A  PROBLEM  FOR  ORIGINAL  STUDY 

FOR  general  directions  see  Chapter  XI. 
The  Models.  —  The  rectangular  block  is  4"  square  by  8" 
long.  It  may  be  made  of  cardboard,  cut  as  in  the  diagram 
(Fig.  74),  and  glued,^  like  the  cube  in  Chapter  XVI.  Or  two 
cubes,  made  as  there  directed,  may  be  used  in  its  place.  The 
cylinder  is  4"  by  8",  and  has  a  circle  about  its  middle. 


_-/6,» 

<■ 

-• ^ 

t 

i 
si 

,-.! 

Fig.  74 

Fig.  75 

Positions.  —  The  block  lies  on  one  long  face,  its  long  edges  par- 
allel with  the  picture  plane.  The  cylinder  stands  on  one  base  in 
front  of  the  block,  touching  it  at  its  middle  (Fig.  75)  The 
models  rest  on  a  surface  three  times  the  height  of  the  block  below 
the  eye,  and  are  four  feet  distant. 

1  The   light  lines  indicate   where  it  is  scored  and  bent  for  the  edges  of  the  block.     The 
quarter-inch  projections  are  laps  for  fastening. 

48 


Chapter  XVI 


THE    FURTHER    STUDY   OF   STRAIGHT- 
LINE  OBJECTS  — A  CUBE  AT  ANGLES 
WITH  THE  PICTURE  PLANE 

THE  Model.  —  For  this  study  make  a  cube,  four  inches  on  a 
side,  from  cardboard  cut  as  in  the  ilhistration  (A,  Fig.  76). 
Pass  a  string  under  one  edge  and  out  of  adjacent  corners 
(B,  Fig.  76)  before  glueing  together. 

Study  of  the  Subject.  —  Turn  the  cube  SO  the  string  comes  from 
the  upper  front  corners,  and  place  it 
as  the  book  was  placed  in  Chapter 
XII  (Fig.  77).  Now,  holding  a  front 
corner  of  the  cube  firmly,  revolve 
the  cube  on  that  corner,  bringing 
the  side  x  into  sight  (Fig.  78).  The 
moment  the  cube  begins  to  revolve, 


A 


16;^- 


the   front,    ^, 


begins 


to  be  turned 
away,  ceasing  to  be 
parallel  with  the  pic- 
ture plane,  and  tend- 
ing toward  coinciding 
with  the  direction  of  seeing 


:^i 


Fig.  76 


In  proportion  as  it 
is  turned  away,  its  right  edge  (H)  becomes  shorter, 
so  that  its  upper  and  lower  edges  (E  and  F)  appear 
to  converge.  The  cube  may  be  revolved  until 
these  edges  (and  their  parallel,  D)  in  their  turn 
converge  directly  in  frout  (Fig.  79),  as  A  and  B  did 
at  first.  Then  the  side  x^  becoming  parallel  with  the  picture 
plane,  will  in  turn  be  seen  in  its  true  shape,  while  its  top  and 
bottom  edges  appear  horizontal. 

4  49 


Fig.  77 


FREEHAND    PERSPECTIVE 


Fig.  78 


Now  turn  the  cube  slowly  back  to  the   position  of  Fig.  78, 
and  with  the   stnngs   find   the   converging   point   of  A  and   B, 

Figure  80  shows  the  cube  in  this  position, 

and  the  vanishing  of  A,  B,  and  C  by  the  use 

of  three  strings.     The  lines  at  right  angles 

to  them,  which  in  Fig.  79  vanished  directly 

in  front,  here  (in  Fig.  80)  vanish  so  far  to  the 

right   that   the  strings   cannot   reach  their 

vanishing  point. 

If  the  cube  is  now  revolved  in  the  opposite   direction,  this 

vanishing  point  (which  we  may  call  VP2  ^)  again  moves 

inward,  as  seen  in  Fig.  81. 

It  will  be  now  readily  seen  that  though  any  set  of 
parallel  horizontal  lines  (as  A,  B,  and  C)  are  directed 
more  to  the  right  or  left,  according  as  the  cube  is  turned, 

they  are  never  actually  raised 
or  lowered.    Hence  their  van- 
ishing point  does  not  move 
up   or  down,  but   is  always      ^'^-  '^^ 
found    on    the    eye    level.     Wo    may 
therefore  conclude  that  receding  hori- 
zontal  lines   always  vanish  in  the   eye 
level. 

We  also  confirm  what  was  observed 
in  Chapter  XII,  that  parallel  lines  vanish 
to  the  same  point. 

Let  us  now  study  the  effect  on  the 
shape  of  its  faces  of  revolving  the  cube. 
In  Figure  77  the  front  face,  «/,  appears 
in  its  true  shape,  while  lines  at  right 
angles  to  this  face  (as  A  and  B)  vanish 
directly  in  front,  and  the  sides  x  and  y 
are  invisible.  As  the  cube  is  revolved 
(Fig.  78)   so   that  x  comes   into  sight,   so  y  is   turned   away,  or 

1  In  distinction  to  that  already  found.    Vanishing  points  are  numbered  in  order  of  finding. 

50 


Fig.  80 


£yc  Cc^&l 

//         ' 

/ 

/ 


Fig.  81 


STUDY   OF    STRAIGHT-LINE   OBJECTS,   ETC. 


ArjAloT  A  CufiE. 

A.B,AA<oC  ARE. 
TOO/TEEO  Fob 
Tut  WIDin  OF 

X;  AND  D,  E 

ANO  F  ABE 
AJOT  yiEEP 
ENOUOM    FOB 
THE    roRE- 

OF    Y- 


G-CoWECTIeH 

OF  A,  8Y 
CHANGINO  THE 
WIDTH   OFTHa 

THEomecnoN 

OF  THE  VANI/H- 
^  ec  f?ioriT. 


B— Correction 

OF  A,  BY  CMANO- 
IklG  THE  _fuANTOF 
THE  VANI/HINO 

eoosf,  whem  the 
wicth  of  the  ■ 

51  D£.^  1;  FOl/NOTO 
flS    «I&MT. 


Fig.  82 


foreshortened.  As  x  widens,  and  its  horizontal  edges  (A  and  B) 
seem  less  steep,  the  other  side  narrows,  and  its  horizontal  edges 
(E  and  F)  appear  more  steep.  Steep- 
ness of  the  horizontal  edges,  therefore, 
goes  ivith  foreshortened  surfaces.  Good 
judgment  on  this  point  is  very  impor- 
tant, as  the  cube  is  the  basis  for  later 
estimates  of  foreshortened  surfaces. 
For  this  reason  much  space  has 
been  given  to  its  study.  It  should 
be  drawn  with  great  care  till  thor- 
oughly mastered. 

The  Recession  of  Horizontal  Surfaces  Toward  the  Eye  Level.  —  It 
will  be  interesting  here  to  place  several  cubes  in  a  receding  row, 
and  see  how  the  vanishing  lines,  being  all  included  in  one  or 
the  other  of  two  sets,  will  vanish  accordingly  to  one  or  the  other 

of  two  vanishing  points.  Taking  out 
every  second  cube  (Fig.  83),  we  find  the 
vanishing  of  those  left  to  be  unaltered. 
We  also  perceive  that  the  table  on 
which  all  rest  seems  to  rise  as  it  re- 
cedes, apparently  tending  to  vanish  or 
merge  itself  in  the  line  marking  the 
eye  level. 

Looking  at  the  tops  of  the  cubes,  all 
situated  in  one  horizontal  plane,  and  recalling  the  horizontal 
surfaces  in  previous  drawings  (as  the  book  covers  and  the 
cylinder  ends)  we  conclude  that  all  horizontal  surfaces  appear 
to  approach  the  level  of  the  eye  as  they  recede.  This  is  seen 
to  be  true  whether  they  are  below  the  eye  or  above  it.  The 
vertical  distance  between  receding  horizontal  planes  must  ap 
pear  less  as  it  is  farther  from  the  eye,  till  at  an  infinite  dis- 
tance it  would  be  entirely  lost,  and  the  parallel  planes  would 
vanish  in  a  line  (the  eye  level)  as  parallel  lines  vanish  in  a 
point. 

51 


Fig.  83 


FREEHAND    PERSPECTIVE 

The  Eye  Level.  —  The  eye  level,  or  level  of  the  eye,  is  not 
actually  a  line ;  but  a  height,  or  invisible  horizontal  plane,  which 
may  be  said  to  extend  indefinitely.  Thus  if  the  student's  eye  is 
five  feet  above  the  ground,  his  eye  level  passes  through  and 
includes  every  point  at  that  height.  But  as  each  one's  eye  level 
is  "  edge  to"  him,  it  would  (if  visible)  always  appear  to  Mm  as  a 
line,  hence  it  is  always  drawn  as  a  line. 


52 


Chapter  XVII 
THE  CUBE  IN  TWO  DIFFERENT  POSITIONS 

THIS  exercise  should  first  be  drawn  from  the  objects,  and 
then  from  memory,  according  to  the  general  directions 
for  memory  work  (Ch.  XI).  Two  drawings  on  one 
sheet,  showing  the  cube  in  different  positions,  are  to  be  made. 
They  should  be  represented  as  of  the  same  size,  which  may  be 


r--  — 


/ 


Fig.  8-1 

done  by  making  their  nearest  vertical  edges  of  the  same  length 
and  at  the  same  height  on  the  paper,  and  using  the  same  eye 
level  for  both. 

Position  of  Models. —  For  the  first  drawing,  place  the  card- 
board cube  so  that  its  front  faces  are  equally  turned  away,  or 
make  angles  of  forty-five  degrees  with  the  picture  plane  (A  in 

5«i 


FREEHAND    PERSPECTIVE 


Fig.  85,  also  plan).     Notice  that  its  upper  back  corner  will  then 
appear  exactly  behind  the  upper  front  one,  the  vertical  sides 


Fig.  85 

of  equal  width  and  the  side  corners  opposite  each  other  and 
equidistant  from  the  center.  In  the  second  position  the  cube  is 
turned  so  its  right  face  makes  an  angle  of 
sixty  degrees  with  the  picture  plane  (B  in 
Figs.  85  and  86). 

Making  the  Drawing.  —  Fasten  the  paper 
in  its  place  on  the  desk  or  have  its  posi-   vuv^  or  a        plah  of  b 
tion  so  marked  that  it  can  be  accurately  ^'^"  ^^ 

returned  to  the  same  place.    Draw  the  margin  lines  and  lightly 
mark  the  extreme  points  for  the  two  cubes  (Ch.  IV).     Note  that 

in  the  first  position  (A,  Fig.  85)  ^the  cube 
occupies  slightly  more  space,  both  horizon- 
tally and  vertically.  Since  the  cube  is  a 
type  solid  the  lines  in  its  final  rendering 
are  simple  and  firm,  only  varying  slightly 
in  thickness  to  suggest  distance.  Begin 
the  first  cube  with  the  easiest  part,  which  is  its  nearest  vertical 
edge.  This  is  parallel  with  the  picture  plane,  and  so  is  drawn 
in  its  true  position.  As  soon  as  this  line  is  placed  mark  the 
eye  level  (in  this  case  it  falls  off  the  paper)  finding  its  height 
as  directed  for  the  book.     The  numbers  on  the  diagram  (Fig.  87) 

54 


DlAGPAM 
SHcM/lNG     A 
CONVENIENT 
ORDER  FOR 
DRAWING  THE 
LINES  OF  THE 
CUliC,  AMD  OF 
RECTANGULAR 
OBJECTS   IN  GENERAL 

Fig.  87 


THE    CUBE    IN    TWO    DIFFERENT    POSITIONS 


give  the  order  iu  which  not  only  cubes,  but  rectangular  ob- 
jects generally,  should  be  drawn.  Get  the  direction  of  lines 
2,  2  by  pencil  measurement  (Ch.  II)  with  espe- 
cial care,  as  their  meeting  with  the  eye  level 
determines  the  vanishing  points.  Hold  the  pen- 
cil vertically  in  front  of,  or  even  touching  the 
nearest  end  of  line  2  (Fig.  88).  Then  keeping 
it  parallel  with  the  picture  plane 
(that  is,  not  receding  as  the  line 
does,  but  resting  in  an  imaginary 
vertical  plane)  revolve  it  down- 
ward to  the  right  until  it  seems  to  cover  line  2 
(Fig.  89).  Holding  it  thus,  with  the  other  hand 
slip  the  paper  (on  which  the 
drawing  has  been  started)  up 
vertically  behind  it  till  the  pen- 
cil touches  the  upper  end  of 
the  vertical  line  already  drawn, 

and  lies  on 


Fig.  88 


Fig.  89 


the  paper, 
sh  o  win  g 


Fig.  90 


the  direction  line  2  should  take  (Fig.  90). 
(This  puts  the  paper  in  the  position  of 
the  picture  plane.)  Draw  this  first  line  2, 
and  mark  its  vanishing  point  on  the  eye 
level  (VPl).  The  direction  of  the  other 
line  2  could  be  found  in  the  same  way 
but  as  in  this  case  they  make  equal  angles 
with  the  picture  plane,  their  vanishing 
points  will  be  equidistant  from  the  center, 

and  VP2  can  therefore  be  so  located,  and  the  second  line  2  drawn 

to  it.     Lines  3,  3  are  then  drawn  (recalling   that  parallel  lines 

converge  to  the  same  vanishing  point). 

For  lines  4,  4  compare  the  apparent  width  of  a  near  vertical 

face  (A  in  Fig.  91)  with  the  front  vertical  line  (B  in  Fig.  91). 

55 


Fig.  91 


FREEHAND    PERSPECTIVE 


(This  front  line,  being  seen  in  its  actual  position  and  unfore- 
shortened,  is  the  best  for  use  as  a  unit  of  measurement.)  Mark  to 
right  and  left  from  line  1  in  the  drawing  the  proportionate  dis- 
tance so  found  (in  this  case  two  thirds  of  line  1)  and  draw  lines 

4,  4.  From  their  upper  extremities 
draw  lines  5,  5  to  their  respective 
vanishing  points. 

For  the  second  drawing  place  the 

^    ?     '  \  lui  ^^^®   ^^    directed,   and    proceed   as 

^L^^^,^^.'    ill    with   the   first   cube.     In   this   case 


I    "// 


Fig.  92 


yP2  falls  so  far  away  that  it  can- 
not be  shown  in  the  illustration 
(B,  Fig.  85).  But  we  know  that 
it  must  fall  somewhere  in  the  eye 
level.  (It  will  be  so  found  in  the 
illustration,  if  tested.)  At  this  stage  a  string  pinned  to  VP4  will 
aid  in  detecting  errors  of  vanishing,  and  will  also  make  real  the 
fact  that  these  lines  must  vanish  precisely  to  their  oivn  vanishing 
point. 

A  Valuable  Testing  Method.  —  After  this  the  following  far  more 
speedy  and  convenient  method  of  testing  should  be  acquired: 
With  one  eye  closed  hold  the  drawing  close  to  the  eye  level,  and 
turn  it  so  that 
one  set  of  van- 
ishing lines  are 
directed  to  the 
open  eye  (Fig. 
92).  Push  the 
drawing  back 
or  forward  as 
needed  till  the 

eye  occupies  the  place  of  the  vanishing  point  for  the  lines  in 
question.  Now  sight  back  over  this  set  of  converging  lines, 
when  it  will  be  found  that  any  failing  to  properly  vanish  are 
quickly  seen   and   easily  noted   for   correction.     A   little   expe- 

5() 


Fig.  93 


THE   CUBE    IN   TWO    DIFFERENT    POSITIONS 

rieuce  is  needed  to  do  this  successfully,  but  it  is  well  worth 
the  trouble. 

Testing  Before  a  Class.  —  An  impressive  method  of  demonstra- 
ting the  vanishing  of  lines  when  teaching  a  class  is  the  following. 
Draw  a  long  horizontal  line  on  the  blackboard  and  mark  it  "  Eye 
Level."  Tack  each  pupil's  drawing  in  turn  on  the  blackboard  so 
that  the  blackboard  eye  level  coincides  with  the  eye  level  of  the 
pupil's  drawing.  With  a  long  ruler  follow  out  one  of  the  vanish- 
ing lines  (Fig.  93),  and  find  its  vanishing  point  on  the  blackboard 
eye  level.  Holding  the  ruler  at  this  vanishing  point  as  a  pivot, 
swing  it  over  the  other  lines  of  the  set  that  should  vanish  to  that 
point.  The  test  is  convincing,  even  to  children;  and  helps 
greatly  to  form  a  standard  of  accuracy. 

It  should  always  be  remembered,  however,  that  such  measur- 
ing is  only  for  testing^  never  for  drawing  the  lines. 


57 


Chapter  XVIII 


A  BOOK  AT  ANGLES  TO  THE  PICTURE 

PLANE 

THE  student  may  copy  this  example  but  must  in  any  case 
draw  from  a  book  similarly  placed ;   and  finally  make  a 
correct  and  spirited  drawing  of  the  same  from  memory. 
The  position  of  this  book  is  like  that  of  the  last  cube  (Ch. 
XVII).     In  studying  this  position  begin  with  the  book  directly  in 


r 


'«aaya<ii>^DeaaiCfflagia»aas:!Wai/.j; 


»rn*irmj«»n'Wt(i*(-'<^icnr2:3r:iHSH| 


Fig.  94 

front  as  in  Chapter  XII.  Note  the  convergence  of  its  ends ;  then 
turning  it  slowly  into  the  required  position  for  drawing  (Figs.  94 
and  95),  observe  how  the  ends  change  in  their  convergence  and 
how  their  vanishing  point  moves  to  the  right  on  the  eye  level 
as  the  book  is  turned.  Look  also  at  the  long  edges  of  the 
book  and  see  how,  at  the  first  movement  of  revolving  it,  they 

58 


A    BOOK    AT    ANGLES    TO    PICTURE    PLANE 

cease  to  appear  horizontal,  and  vanish  toward  a  point  which, 
though  at  first  infinitely  distant,  must  nevertheless  fall  on  the 
eye  level. 

Drawing  the  Book.  —  Sketch  the  margin  lines,  and  plan  a  good 
position  of  the  book  in  relation  to  the  inclosed  space.  Mark  the 
height  of  the  eye  level  as  soon  as  a  dimension  (as  xy.  Fig.  95)  by 


Fig.  95 

which  it  can  be  estimated  is  decided  on.  Find  the  direction  of  the 
book  edges  (corresponding  to  lines  2,  2  in  the  cube  in  Ch.  XVI) 
with  especial  care.  Sketch  in  the  book  with  delicate  lines,  pro- 
ceeding in  the  order  observed  when  drawing  the  cube,  and 
correcting  where  necessary. 

Artistic  Expression.  —  Finally,  the  subject  should  be  rendered 
artistically.  To  accomplish  this,  the  line  is  adapted  to  the  qual- 
ity of  that  portion  on  which  it  is  used.  Certain  features  may  be 
selected  for  use  to  augment  interest ;  as  the  curving  ridges,  the 
ornament,  and  the  title  space  on  the  back,  or  even  the  worn 
corners.  But  having  expressed  in  these  details  the  point  intended 
(as  a  worn  corner  by  the  shape  of  its  boundary  line)  take  care  to 
do  no  more.  It  is  wearisome,  for  instance,  to  see  lines  on  these 
corners  to  represent  the  separation  into  layers  caused  by  wear. 
Lines  also  produce  a  dark  color,  while  worn  corners  are  generally 
light;  and  are  also  undesirable  places  for  the  use  of  dark 
spots. 

59 


FREEHAND    PERSPECTIVE 

As  the  vertical  edges  of  the  cube  are  drawn  vertical  because 
parallel  with  the  picture  plane,  so  the  corners  of  the  book  must  be 
made  vertical  in  the  drawing,  as  they  are  in  reality.  For  in- 
stance, points  C  and  D  being  in  a  vertical  line,  must  be  so  placed 
in  the  drawing.  The  same  is  true  of  the  curves  on  the  back  of 
the  book. 

At  this  point  the  student  readily  sees  that  all  vertical  lines 
(since  the  picture  plane  is  vertical)  will  he  parallel  to  the  picture 
plane,  and  must  invariably  be  draivn  as  they  actually  are,  or  vertical. 


60 


Chapter  XIX 


TWO  BOOKS  AT  DIFFERENT  ANGLES  TO 
THE  PICTURE   PLANE 


B 


EGIN  the  study  of  this  subject  by  placing  the  books  as 
in  Fig.  97.  Observe  that  in  this  position  there  is  but 
one  vanishing   point  for   the  two  objects,  the   ends   of 


Fig.  96 


the  books  being  all  parallel,  and  their  other  horizontal  edges 
parallel  with  the  picture  plane.  Now  turn  the  whole  group, 
as   in   Fig.   98,   and    see   that   we   have    two   vanishing   points, 

61 


FREEHAND    PERSPECTIVE 


EYiUvU- 


one  for  the  ends  and  the  other  for  the  long  edges  of  the 
books. 

Now  revolve  the  upper  book  a  little  more  (Fig.  99),  so 
that  its  horizontal  edges  cease  to  be  parallel 
to  those  of  the  other, 
and  it  will  have  its 
own  points  of  con- 
vergence (VPS  and  ^-^vf 
VP4).  Its  length  ap-  ^ 
pears  lessened,  and  its 
ends  longer,  for  this  Fig.  98 

change.  The  shortened  edges  vanish  more  steeply,  and  those 
which  have  become  longer  appear  less  steep.  We  find,  as 
would  be  expected,  that  when  Imes  cease  to  he  parallel,  their 
vanishing  points  are  different. 


Fig.  97 


•*•  TjCf A J<, 


Fig.  99 


62 


Chapter  XX 


THE  ACTUAL  CENTER  OF  THE  CIRCLE  AND 
MEASUREMENT  INTO  THE  PICTURE  BY 
PARALLEL  LINES 

PRELIMINARY  Study.  —  Does  the  eye  see  half  way  round  the 
cylinder  I     The  question  is  best  answered  by  experiment. 
Holding   the   cylinder  vertically  and  rather   near  (to 
more  easily  see  the  facts),  mark  on  it  the  points  where  the  side 
boundaries  appear  to  meet  the  top  (A  and  B  in  Fig.  101).     It  will 


Fig.  100 

be  found  that  they  are  actually  less  than  half  way  from  the  front 
to  the  back  (Fig.  102).  Yet  the  pencil  has  marked  what  the  eye 
mw  as  the  greatest  dimension.  As  shown  in  Fig.  103  this  appar- 
ent greatest  dimension  (A  B)  forms  the  long  diameter  of  the  ellipse 

63 


FREEHAND     PERSPECTIVE 


Showing  apparent  middue  from 

FffONT  To  BACK.  OR  LONG  DIAMETER 

Fig.  101 


in  the  perspective  view.    It  is  evident,  therefore,  that  the  eye  does 
not  see  half  way  round  the  cylinder,  and  (as  seen  in  Ch.  IV)  that 

the  long  diameter  of  the  ellipse  is 
not  an  actual  diameter  of  the  circle, 
while  that  portion  of  the  circumfer- 
ence beyond  the  long  diameter  (A  B) 
is  actually  more  than  half  of  the  circle, 
the  part  in  front  of  A B appearing  equal 
to  it  only  because  nearer  to  the  eye. 

The  actual  position  of  the  apparent 
greatest  dimension  (the  long  diameter) 
changes  with  the  position  of  the  observer. 
The  plan  (Fig.  102)  shows  that  C  D  would  appear  as  the  greatest 
dimension  if  the  eye  should  be  at  2.  This  also  may  be  seen  by 
experiment  (as  in  Fig.  101). 

Planning  the  Exercise.  —  In  placing  this  exercise  observe  that 
the  perspective  of  the  concentric  square  and  circles  is  made 
much  larger  than  the  geometric  diagram, 
to  show  more  clearly  the  perspective 
details. 

Drawing  the  Circles.  —  When  the  square 
has  been  drawn  in  perspective  (like  the 
top  of  the  cube  in  Fig.  77)  its  actual 
center  (o  in  Fig.  103)  is  found  at  the 
crossing  of  its  diagonals,  as  in  the  geo- 
metric diagram  above.  In  the  diagram 
the  ends  of  its  diameters  mark  the  points 
(C,  D,  E,  and  F)  where  the  circle  touches 
the  square,  and  they  will  do  the  same  in 
the  perspective.  The  diameters  pass  through  the  true  center  and 
one  is  parallel  to  the  picture  plane.  It  can  therefore  be  drawn 
in  the  perspective  in  its  actual  direction,  giving  two  points 
(c  and  d).  The  other  diameter,  being  parallel  with  the  receding 
sides  of  the  square,  vanishes  with  them  in  the  eye  level  directly 
in  front  (at  VPl)  giving  points  e  and  /.     Now,  though  the  actual 

61 


R-ANOFABCVC, 
5HO"0t/lN&  THE 

ACTyAu  Place 

OF  THE   LON& 
DIAMETER,  A  B 


ACTUAL  CENTER  OF  CIRCLE,  ETC. 


Geometric    diagram. 


diameter  of  the  circle  touches  the  square  in  c  and  d,  the  ellipse 

appears  longest  at  a  part  nearer  than  c  and  d,  which  seems  to 

be  exactly  half  way  between   e  and  /.     Through  this   half-way 

point  {x)  the  long  diameter   can 

be  drawn ;  making  it  longer  than 

c-d,  and  yet   not  quite   touching 

the   square.     The   ellipse   is  then 

easily  sketched  through  these  six 

points  (a,  6,  c,  d,  e,  and/),  making 

it  symmetrical  on  a-b  and  e-f. 

For  the  other  ellipses  the  points 
where  they  cross  the  actual  diam- 
eter of  the  circle  [c-d)  are  marked 
by  lines  from  1,  2,  3,  and  4  which 
vanish  in  VPl,  giving  four  points 
(9,  10,  11  and  12),  two  for  each 
of  the  smaller  ellipses. 

Measuring  Distances  into  the 
Picture.  —  For  the  front  and  back 
points  of  these  ellipses,  line  ef 
must  be  divided  into  six  perspec- 
tivelij  equal  parts,  as  EF  in  the 
diagram  is  divided  into  six  actu- 
ally equal  parts.  This  can  be  done,  and  in  practice  usually  is 
done,  by  the  eye  (as  for  the  cylinder  in  Ch.  IV),  noting  that 
the  true  center  (o),  already  known,  is  one  point  of  division.  But 
the  use  of  the  diagonal  for  such  distances  is  simple  and  often 
a  convenience.  Thus  it  is  easy  to  see  that  in  the  diagram  the 
vertical  lines  from  1,  2,  3,  and  4  cut  the  diagonals  proportionately 
to  the  divisions  on  GH^  in  this  case  into  six  equal  parts.  These 
divisions  can  in  turn  be  transferred  to  EF  by  horizontal  lines 
from  the  points  on  the  diagonal  HI,  giving  5,  6,  7,  and  8,  the  four 


1  Students  of  geometry  will  recognize  in  this  the  problem 
of  dividing  a  line  proportionately  by  means  of  parallel  lines  cross- 
ing a  triangle. 

5  65 


Perspective  representation  of  above. 
Fig.  103 


Fig.  104 


FREEHAND     PERSPECTIVE 

points  needed.  In  the  perspective  the  method  is  the  same,  using 
lines  perspectively  parallel  to  e-f — that  is,  the  lines  already 
drawn  from  1,  2,  3,  and  4  to  VPl.  This  use  of  the  diagonal 
occurs  further  on,  as  for  the  steps  in  Chapter  XXII. 

A  Second  Method.  —  The  vanishing  point  of  the  diagonal  can 
also  be  used  to  obtain  these  points.  Thus  the  diagram  shows 
that  lines  from  1,  2,  9,  and  10  parallel  with  the  diagonal  GJ  will 
mark  on  EF  the  same  divisions.  In  the  perspective  these  lines 
will  appear  perspectively  parallel  to  the  diagonal  —  that  is,  drawn 
to  the  same  vanishing  point.  Since  they  are  horizontal,  that 
vanishing  point  will  be  on  the  eye  level.  Therefore  the  diagonal 
G-J  can  be  carried  out  to  the  eye  level  to  find  its  vanishing  point 
(VP2)  to  which  the  parallel  lines  are  drawn. 

The  principle  to  be  remembered  for  use  is:  Whatever  meas- 
urements can  he  obtained  geometrically  by  the  use  of  actually  parallel 
lines,  can  be  obtained  in  perspective  by  the  use  of  perspectively  parallel 
lines. 

It  must,  however,  be  noted  that  these  are  only  relative  meas- 
urements. A  first  distance  into  the  picture  —  the  foreshortened 
width  of  the  square  in  this  case  —  is  determined  freehand  by 
past  experience  (as  with  the  cube).  Mechanical  perspective 
gives  methods  of  obtaining  this  first  distance,  the  position  of  the 
eye  and  the  picture  plane  being  given.  It  can  also  be  obtained 
from  a  side  view,  by  using  the  same  data.  Both  these  methods 
are  too  complicated  for  common  use  in  freehand  work.  Such 
proportions  are  so  easily  estimated  by  recalling  the  cube  that  it 
is  better  to  rely  on  a  trained  judgment  for  them. 


66 


Chapter  XXI 
BOOKS  WITH  A  CYLINDRICAL  OBJECT 


T 


HE  student  should  take  this  exercise  as  previous  ones, 
copying  first  if  he  needs  to  do  so,  then  composing  and 
sketching  a  similar  study,  and  finally  making  a  drawing 


Fig.  105 


from  memory.  For  both  of  the  latter  several  different  arrange- 
ments of  objects,  with  trial  sketches,  should  be  made;  and  the 
best  chosen  to  use  in  the  final  drawing. 

The   Finder   (Ch.   VIII)    should    be    used    to    compare    the 
effect  of  different  compositions,  also  the  effect  of   cutting  out 

67 


FREEHAND    PERSPECTIVE 

compositions    from    larger    ones    by    different    margins.      Note 
that  in  Fig.  106,  with  a  tall  object,  the  books  are   turned  so 

that  their  horizontal  dimensions  are 
not  great  enough  to  neutralize  the 
dominant  vertical  effect.  In  Fig. 
107,  on  the  other  hand,  the  long 
horizontal  dimensions  of  the  books 
and  the  low  flat  dish  harmonize 
very  well,    and    this    arrangement 


Fig.  106 


Fig.  107 


necessitates  a  marginal  rectangle  longer  from  side  to  side. 
The  books  and  dish  alone  make  a  good  simple  arrangement, 
but  the  tray  may  be  added  if  desired. 


Fig.  108 


68 


Chapter  XXII 
THE  STUDY  AND  DRAWING  OF  A  HOUSE 


M 


ODEL  for  the  Study.  —  Make  an  equilateral  triangular 
prism  from  cardboard  cut  as  in  the  diagram  (Fig.  110), 
and  place  it  on  the  top  of  two  cubes.     Put  a  box  or 


Fig.  109 

books  on  the  table  under  this  model,  raising  it  so  that  the  level 
of  the  eye  will  fall  one-fourth  way  up  on  the  cubes  (Fig.  Ill) . 
Place  the  model  about  sixteen  inches  from  the  eye,  and  turn  it 
so  its  long  edges  will  make  angles  of  thirty  degrees  with  the  pic- 
ture plane  (Fig.  119) .  It  may  now  be  regarded  as  the  type  form 
of  a  house,  seen  (in  proportion  to  its  size)  from  an  ordinary  posi- 

69 


FREEHAND    PERSPECTIVE 


8!4-^ 


Fig.  110 


tion  for  viewing  a  house.     By  aid  of  the  imagination,  it  may- 
be regarded  as  a  house  of  two  stories,  with  a  front  door  in  the 

middle  of  a  side,  the  box  top  taking 
the  place  of  the  ground. 

This  exercise  should  be  first  drawn 
in  thin,  light  lines,  studying  the  dia- 
grams, and  following  the  directions. 
The  construction  lines  should  then  be 
erased,  and  the  drawing  rendered  as 
shown  in  Fig.  109. 

Drawing  the  Exercise.  —  Begin  with 
the  nearest  vertical  edge  of  the  house. 
The  model  was  placed  so  that  the 
level  of  the  eye  should  be  one-fourth  way  up  the  height  of  its 
rectangular  part  because  the  eyes  of  a  person  standing  might 
be  about  five  feet  above  the 
ground,  and  the  height  of  a 
two-story  house  at  its  eaves 
about  twenty  feet.  Mark  the 
eye  level  on  the  paper  there- 
fore, one  fourth  of  the  height 
of  the  nearest  edge  from  its 
bottom,  and  take  the  direction 
of  lines  A  and  B  (Fig.  112) 
to  determine  the  two  vanishing  points,  exactly  as  was  done 
with  lines  2,  2  in  the  cube  (Ch.  XVII)  although,  being  above 
the  eye,  they  appear  to  tend  downward.  Draw  the  two  lower 
horizontal  lines  and  the  side  vertical  lines  as  those  of  the  cube 
were  drawn. 

To  construct  the  roof  recall  that  the  end  of  our  model  is 
an  equilateral  triangle  (Fig.  113)  with  its  apex  over  the  center 
of  the  house  end.  Draw  the  diagonals  of  this  square  house  end 
and  carry  up  a  vertical  line  of  indefinite  length  from  its  center, 
on  which  the  apex  of  the  gable  is  to  be  marked.  The  actual 
roof  height  of  our  small  model  may  be  found  by  this  diagram ; 

70 


Fig.  Ill 


THE    STUDY    AND    DRAWING    OF    A    HOUSE 

but  as  that   makes  a  roof  steeper  than  is  usual,  we  will  set  it 
off  less  in  the  drawing,  that  is,  making  EC  (Fig.  112)  but  a 


Fig.  112 

little  more  than  CD.  (Since  these  distances  are  in  the  same 
vertical  line,  and  so  at  the  same  distance  from  the  picture  plane 
they  are  seen  and  drawn  in  their  true  proportions  to  each 
other.) 

The  sloping  sides  of  the  gable  may  then  be  drawn 
to  the  house  corners,  and  the  ridgepole  to  VP2.  The 
gable  apex  on  the  other  end  may  be  found  by  drawing 
a  vertical  line  from  the  center  (x)  of  the  invisible  end 
to  cut  the  ridgepole.  Its  slanting  sides  are  drawn  to 
complete  the  blocking-in  lines  thus  far  of  the  house. 

71 


c ^ 


E.ND  View 
Fig.  113 


FREEHAND    PERSPECTIVE 


Oblique  Vanishing  Lines.  —  We  have  said  little  about  the  slop- 
ing end  lines  of  the  roof.  But  now,  looking  again  at  our 
model  (Fig.  Ill),  we  see  that  the  ridgepole,  R,  because  it  is 
farther  away  than  the  eaves,  appears  shorter,  so  that  the 
slanting  ends  (F  and  Gr)  of  the  front  surface  of  the  roof  appear 
to  converge  upward.  Turning  to  the  drawing,  we  find  in  con- 
firmation of  this  that  (if  the  drawing  has  been  carefully  made) 
these  lines  do  thus  converge.  Now  let  us  search  for  the  general 
truth  governing  that  convergence.  These  slanting  ends  are  not 
horizontal,  so  that  we  should  not  expect  them  to  tend  toward  the 
eye  level ;  and  we  observe  that  they  do  not.  But  they  are  actu- 
ally parallel  to  each  other  and  therefore  must  vanish  or  appear 
to  converge  to  the  same  point.  Hoiv  to  find  that  point  is  the 
question. 

Put  some  books  in  the  place  of  the  house  model,  and  arrange 
them  so  that  their  edges  vanish  like  the  house  edges.     Now  raise 

the  upper  cover  (A, 
^''  ~  Fig.  114),  and  observe 

that  its  ends  1  and  2, 
though  still  parallel 
with  each  other  like 
the  ends  of  the  roof, 
have  ceased  to  be 
parallel  with  the 
other  book  ends. 
They  therefore  no 
longer  vanish  toward 
VPl,  but  to  a  higher 
point.  We  have  not, 
however,  turned  these 
edges  to  right  or  left,  but  have  simply  lifted  their  farther  ends 
or  revolved  them  in  parallel  vertical  planes.  Therefore  their 
vanishing  point  cannot  move  to  right  or  left;  but  as  they  are 
revolved,  must  appear  to  move  directly  uptvard,  or  in  a  vertical  line 
passing  through  VPl.     This  continues  until  the  cover  becomes 

72 


Fig.  114 


THE    STUDY    AND    DRAWING    OF    A    HOUSE 


vertical;  when  its  ends  appear  in  their  true  position  and  cease 
to  vanish,  like  all  vertical  lines. 

Place  strings  under  the  cover,  as  in  Chapter  XII.  Holding 
the  strings  with  the  left  hand  as  in  the  illustration  (B  in  Fig. 
114)  raise  the  cover  with  the  right  (keeping  the  strings  parallel 
to  the  picture  plane).  By  this  experiment  their  convergence 
toward  a  point  in  the  vertical  line  from  VPl  is  more  plainly 
shown.  Since  these  slanting  book  ends  are  neither  horizontal 
nor  vertical  but  oblique  to  both  directions,  their  vanishing  point 
or  that  of  any  set  of  oblique  lines,  may  be  distinguished  as  an 
Oblique  Vanishing  Point,  or  OVP. 

By  revolving  the  upper  cover  farther,  or  opening  the  lower 
cover  and  using  the  string  (A  in  Fig.  115),  oblique  vanishing 
points  below  the 
eye  level  may  be 
determined.  And 
the  apparent  di- 
rection of  oblique 
lines  can  be  found 
with  the  pencil  ex- 
actly as  that  of 
any  line  (B  in 
Fig.  115). 

Vanishing  Traces. 
—  By  turning  one 
of  these  illustra- 
tions around,  to  bring  the  eye  level  vertical,  as  in  Fig.  116,  it 
will  be  seen  that  the  line  containing  OVPl  and  0VP2  serves 
a  purpose  similar  to  that  of  the  eye  level.  We  note  that 
the  surface  formed  l)y  the  visible  ends  of  the  hooJcs  appears  to 
recede,  or  vanishes,  toward  this  line.  If  a  larger  book  be  placed 
against  the  other  ends,  the  surface  of  the  larger  book,  being 
parallel  to  the  visible  ends  of  the  other  books,  will  be  found  to 
vanish  toward  the  same  line.  It  may  be  concluded  that  all  sur- 
faces parallel  to  the  hook  ends  in  this  case  will  vanish  in  this  line, 

73 


Fig.  115 


FREEHAND    PERSPECTIVE 


exactly  as  all  horizontal  surfaces  appear  to  vanish  toward  the 
eye  level.  We  may  call  this  line  a  Vanishing  Trace.  The  eye 
level  is  such  a  vanishing  trace  for  all  horizontal  surfaces.     See 

note,  Chapter  XI,  in  Solu- 
tions of  Problems. 

First  replacing  the  house 
model  as  in  Fig.  Ill,  we  now 
turn  to  the  drawing  and  test 
these  oblique  lines  (P  and  Gr 
in  Fig.  112).  If  correctly 
drawn  they  will  be  found 
to  converge  toward  a  point 
(OVPl)  directly  above  VPl. 
At  once  use  is  made  of  this 
point  for  drawing  the  ends 
of  the  roof  projection  (sug- 
gested in  the  model  by  pin- 
ning cardboard  as  in  Fig. 
117).  These  edges  are  par- 
allel with  the  corresponding  roof  edges,  like  the  book  margins; 
so  their  width  can  be  set  off  on  the  upper  line  of  the  house  (B, 
Figs.  112  and  117)  to  right  and  left, 
(points  X  and  y)  remembering  that 
the  nearer  distance  appears  a  little 
greater.  Through  these  points  draw  --^ 
lines  vanishing  to  OVPl.  A  similar 
projection  is  measured  downward  on  a 
continuation  of  the  oblique  gable  edge 
F  beyond  its  lower  end  (z) ;  and  through 
this  point  a  line  parallel  to  B  (that  is,  vanishing  with  it  in  VP2) 
forms  the  eaves.  (The  estimation  of  these  last  measurements  by 
the  eye  forms  an  important  part  of  the  student's  training  and 
should  be  carefully  thought  out.  Thus  the  eaves  projection  from 
line  B  forward  is  more  foreshortened  than  the  gable  projection 
from  F  to  the  left;  and  distances  should  be  set  off  accordingly.) 

74 


Fig.  116 


I  E 


Fig.  117 


THE    STUDY    AND    DRAWING    OF    A    HOUSE 


BAC»\   VIEW 


End  View 


Fig.  118 


The  lower  oblique  vanishing  point  (0VP2)  is  used  for  the 
projections  on  the  back  slope  of  the  roof.  Continue  the  back 
oblique  roof  edge  (line  H)  to  meet  the  vanishing  trace  through 
VPl,  giving  0VP2.  Draw  line  K  from  the  near  end  (I)  of  the 
ridgepole  to  0VP2,  and  cut  it  by  a  line  from  the  nearest  eaves 
corner  to  VPl.  Where  this  line  cuts  the  oblique  edge  K  will  be 
the  eaves  corner  (M)  for  the  far  side  of  the  roof ;  and  a  line  from 
it  to  VP2  fornas  the  eaves  on  that  side. 

There  is  another  way  of  getting  the  projections  on  the 
further  slope  of  the  roof,  which  is  useful  in  case  0VP2  falls  too 
far  away  to  be  conveniently  used. 
Turning  the  model  we  see  that  the 
invisible  line  (L  in  back  view,  Fig. 
118),  if  carried  to  the  edge,  ends 
in  O,  horizontall}^  opposite  x  (end 
view).  A  line  from  x  through  O 
would  therefore  vanish  in  VPl. 
Hence,  to  obtain  0,  line  L  is  carried  forward  indefinitely,  and 
cut  by  a  line  from  x  to  VPl.  The  desired  edge  is  then  drawn 
from  point  I  through  O  indefinitely,  and  cut  by  a  line  from  J  to 
VPl,  giving  the  corner,  M. 

The  "L"  Part  of  the  House.  —  The  plan  (Fig.  119)  shows  its 
position.     Its   width   is   marked   off   on   the   farther   (invisible) 

end  of  the  house,  and  it  is  drawn 
as  was  the  main  part  of  the 
house.  Note  the  less  steep  slope 
of  the  porch  roof  (Fig.  112),  so 
that  its  oblique  lines  are  not 
parallel  with  those  of  the  other 
roofs  but  have  another  vanishing 
point,  0VP3,  lower  in  the  same 
vertical  line. 

Windows  and  Doors.  —  The  windows  and  doors  may  be  marked 
on  the  model  (Fig.  120).  It  will  be  easily  seen  that  their  top 
and  bottom  edges  are  all  parallel  to  the  horizontal  lines  of  the 

75 


Fig.  119 


FREEHAND    PERSPECTIVE 


side  on  which  they  are  located  and  therefore  vanish  to  the  same 
point.  Mark  their  heights  on  the  nearest  vertical  edge  of  the 
house,  and  draw  lines  (as  P,  in  Fig.  112),  thence  to  the  vanishing 


-D- 


DD  D  DD 


fEONT  VIEW 

Fig.  120 


points.  On  these  lines  their  perspec- 
tive widths  are  to  be  set  off.  Begin 
with  the  door.  Find  the  middle  of 
the  house  front  by  its  diagonals,  and 
make  the  near  half  of  the  door  a 
little  wider  than  the  far  one.  Check 
this  by  seeing  that  the  remaining 
distances  (from  the  door  to  the  front  corners  of  the  house)  are 
also  perspectively  equal,  that  is,  the  near  one  larger.  Mark  the 
sides  of  the  windows  in  the  same  way.  Remember  that  since 
the  space  between  the  near  window  and  the  near  corner  is  con- 
siderably nearer  to  us  than  that  between  the  far  window  and  the 
far  corner,  more  difference  should  be  made  in  their  size  than 
between  the  halves  of  the  door. 

The  width  of  the  windows  on  the  end  of  the  house  should  be 
to  that  of  the  front  ones  as  the  right  face  of  the  second  cube  in 
Chapter  XVII  (Fig.  84)  is  to  the  right  one.  The  height  of  the 
windows  in  the  "  L "  is  made  the 
same  perspectively  by  carrying  their 
measurements  from  the  right  front 
corner  of  the  main  house  on  lines 
vanishing  to  VPl.  These  lines  lie 
on  the  invisible  end  of  the  main 
house,  and  from  where  they  reach  the 
"L"  are  continued  along  its  front  by 
lines  running  to  VP2. 

The  Chimney.  —  To  better  visualize  this  part  of  the  house,  cut 
and  fold  cardboard  as  in  Fig.  121.  Get  the  slope  of  lines  1,  2,  3, 
and  4  by  laying  the  cardboard  against  the  apex  of  the  gable 
and  marking  around  it.  Stand  this  model  on  the  roof  in  its 
middle,  and  after  marking  on  the  roof  around  it,  cut  out  the 
space  so  marked  and  push  the  chimney  down  through  the  open- 

76 


--4-  IN 


Fig.  121 


THE    STUDY    AND    DRAWING    OF    A    HOUSE 

iug  until  it  projects  the  proper  distance  above  the  roof.  Lay  a 
pencil  on  the  roof  against  the  chimney  (Fig.  122) ,  and  move  it  to 
the  left  without  changing  its  direction  till  it  coincides  with  the 
gable  edge.  This  shows  the  gable 
edge  and  the  oblique  line  where  the 
chimney  passes  through  the  roof  to 
be  actually  parallel.  This  oblique 
line,  therefore,  has  the  same  vanish- 
ing point  as  that  of  the  gable  line, 
which  is  OVPl.  The  top  of  the 
chimney  front  and  the  line  below 
it  (AB,  Fig.  123)  are  parallel  to  the 
eaves  and  ridgepole.  Turn  the  model  (Fig.  120,  end  view)  and 
see  that  the  top  edge  of  the  chimney  is  parallel  with  the  hori- 
zontal lines  on  the  house  end,  which  we  have  already  drawn ^ to 


Fig.  122 


Fig.  123 

VPl.  A  pencil  held  horizontally  and  moved  slowly  up  in  front 
of  the  model  will  help  to  see  this  as  will  marking  the  chimney  in 
the  model  off  into  bricks  (Fig.  124). 

Drawing  the  Chimney.  —  Continue  on  the  roof  the  center  line 
used  for  the  door  (that  is,  vanish  a  line  from  its  top  to  OVPl), 

77 


^ 


FREEHAND    PERSTECTIVE 


r^w^ra^s^ 


a!& 


J 


A-  Plan  or  Cmimney. 
g-  "Profile    smowino  orna- 
mental. BAND  AT  TOP. 

Fig.  124 


and  mark  down  from  the  ridgepole  on  this  line  half  the  thickness 
of  the  chimney  (judged  by  the  eye).     Draw  a  line  through  this 

point  toward  YP2,  and  on  it  set  off  to  right 
and  left  perspectively  equal   distances   for 
the  breadth  of   the   chimney  (AB).     Draw 
line  C  to  OVPl.    Where  it  crosses  the  ridge- 
pole (D)  is  the  middle  of  the  chimney  from 
front  to  back.     Make  the  far   half   of  the 
chimney  proportionately  as   much   smaller 
than  the  near  half   as  the  far  half  of   the 
house  end  is  smaller  than  its  near  half. 
The  projecting  band  at  the  top  of  the  chimney  is  shown  in 
plan  and  profile  in  Fig.  124.     Its  perspective  is  drawn  as  are 
projecting  book  covers.     Be  careful  to  represent  the  backward 
projection  on  the  farther  side. 

The  Steps.  —  For  these  the  detail  drawing  (Fig.  125)  is  first 
made.  The  height  under  the  threshold  of  the  door  is  a  little 
less  than  two  feet,  or  about 
one  third  of  the  height  of 
the  eye  —  enough  for  three 
steps.  Divide  the  vertical 
line  under  the  near  edge  of 
the  door  therefore  into  three 
equal  parts,  and  draw  lines 
of  indefinite  length  to  VPl 
through  the  four  points  of 
division.  On  the  lower  line, 
B,  mark  off  the  proper  dis- 
tance (as  four  feet),  which 

may  be  estimated  by  comparison  with  the  windows  on  the  end 
of  the  house  (their  width  being  parallel  with  these  lines,  and 
usually  about  three  feet).  Divide  this  distance  into  perspective 
halves.  A  vertical  line  from  the  near  end  of  line  B,  cutting  line 
E  in  Point  2,  completes  the  rectangle,  1-2-3-4,  the  middle  of 
which   can  be  found  by  its   diagonals,  giving  the   perspective 

78 


THE    STUDY    AND    DRAWING    OF    A    HOUSE 


^ 

\^ 

c> 

\/ 

<» 

.^B 

/■^ 

B 


Fig.  126 


halves  required.     (See  A,  Fig.  126.)     The  further  half  is  for  the 
wide  top  step.     The  near  half   of  the  rectangle  can  be  divided 
again  in  the  same  way  for  the  two   lower  steps.     Where  the 
vertical  line  from   the  near  end  of   B  cuts 
line  C  is  the  upper  near  corner  of  the  lower 
step.    A  vertical  line  through  O  will  mark  its 
width  on  C,  and  continued  to  cut  D  forms 
the  nearest  front  corner  of  the  second  step. 

Another  method  of  sketching  the  steps 
is  shown  in  Fig.  126.  When  the  first  step 
has  been  drawn  its  diagonal  is  continued 
through  6,  cutting  line  D  in  7,  and  forming 
the  diagonal  for  the  second  step,  which  is  completed  by  con- 
tinuing line  D  to  cut  a  vertical  from  6  in  8.  This  can  be  con- 
tinued for  as  many  steps  as  needed.  The  diagonal  can  also  be 
used  as  a  test  for  steps  drawn  by  the  first  method. 

The  long  edges  of  the  steps  vanish  in  VP2,  and  are  cut  alter- 
nately by  lines  vanishing  in  VPl  and  vertical  lines. 

The  Dormer  "Window.  —  This  is   constructed  in   principle  like 
the  gable  of  the  roof.     The  detail  drawing  (Fig.  127)  should  be 

carefully  studied,  and  drawn  sepa- 
rately if  desired,  before  sketching 
the  window  on  the  house. 

On  the  center  vertical  line  of  the 
house  front  continued  upward,  mark 
the  height  of  the  dormer  from  line 
B.  (In  this  case  it  is  not  so  high 
as  the  main  house.)  Through  this 
point  (S)  the  dormer  ridgepole  is 
drawn  to  VPl,  and  cut  by  the 
oblique  middle  line  on  the  roof  (in  point  T).  The  width  (1-2) 
of  the  dormer  is  then  marked  perspectively  to  right  and  left  on 
line  B.  Through  these  points  (1  and  2)  the  "  valleys,"  or  meeting 
lines  of  the  dormer  with  the  main  roof,  are  drawn  from  the  roof 
end  (T)  of  the  dormer  ridgepole  to  the  edge  of  the  eaves  (points 

79 


Fig.  127 


FREEHAND    PERSPECTIVE 

U  and  V) .  From  these  points  the  edges  of  the  dormer  roof  pro- 
jection  run  parallel  respectively  to  A  and  C. 

Oblique  Lines  in  the  Dormer.  —  These  two  lines  A  and  C, 
though  oblique  to  line  B,  are  in  the  same  vertical  plane  (as  the 
gable  lines  F  and  H  in  the  main  house  are  in  the  same  plane 
with  line  A  in  Figs.  Ill  and  112).  Therefore  draw  a  second 
vanishing  trace  for  oblique  lines  vertically  through  VP2,  and 
continue  A  upward  until  it  cuts  this  trace  in  0VP4,  to  which 
draw  the  edge  D.  The  other  oblique  edge  (E)  vanishes  in  the 
same  vertical  below  VP2. 

When  experience  has  been  acquired,  such  oblique  lines  can 
be  satisfactorily  drawn  without  actually  finding  their  vanishing 
points.  Such  convergences  are  generally  estimated  in  practical 
work.  But  estimates  are  much  more  valuable  when  made 
with  a  knowledge  of  methods  by  which  they  can  be  definitely 
determined. 


80 


Chapter  XXIII 


A  BUILDING  FROM  THE  PHOTOGRAPH  OR 

A  PRINT 


T 


HE  example  given  in  Fig.  128  is  from  the  old  church  of 
San'  Apollinare  in  Classe,  near  Ravenna. 

The  beginner  may  draw  this  as  a  preparation  for  his 


Fig.  128 

next  work,  which  should  be  the  drawing  of  a  building  or  part  of 
one  from  a  print  of  his  own  selection. 

Making  a  Selection.  —  This  choice  should  be  carefully  made, 
care  being  taken  to  secure  unity,  or  an  appearance  of  one  whole 

6  81 


FREEHAND    PERSPECTIVE 


Fig.  129 


thing   having  a  center  of  interest  and   parts  which  are  subor- 
dinated, or  catch  the  eye  less  quickly.     It  should  be  well  placed 

in  its  rectangle  (Chs.  VIII  and  XXI). 
For  instance,  the  tall  tower  in  Fig.  129 
needs  a  margin  that  is  longest  vertically, 
and  quite  narrow,  to  produce  a  harmony 
of  lines.  The  smaller  buildings  with  it 
give  variety,  and  by  a  contrast  which  is 
not  too  great  emphasize  its  height,  be- 
ing subordinated  that  the  tower  may 
remain  prominent  in  the  composition. 

In  Fig.  130,  on  the  other  hand, 
the  long, 
low  mass 
of  farm 
buildings 
set  well 
back  into 
the  pic- 
ture re- 
quires a  rectangle  that  is  longer  horizontally. 

Some   of  the   different  selections  that  may  be  made  from 

one  print  (Fig.  131)  are 
shown  in  Fig.  132.  The 
beginner  can  by  such 
means  obtain  an  example 
simple  enough  to  be  with- 
in his  powers  and  often 
a  better  composition. 

Drawing  from  the  Print. 
—  As  soon  as  the  place 
of  the  building  on  the 
paper  is  fixed,  the  Jeml 
of  the  eye  must  he  deter' 
mined  and  marked,  and  the  vanishing  points  of  the  principal  sets 


'H'iTi 


Fig.  130 


^■■^■^■KT^^ 

[                ^'^%           ffi               JVMBKl 

1 

f  ■ 

k    "1 

BHHr 

^^^K|s£|^|u^^ 

Fig.  131 


BUILDING    FROM    PHOTOGRAPH,    ETC. 


of  horizontal  lines  must  he  found  on  that.     It  is  of  course  easier 

for  the  beginner  to  use  such  vanishing  points  as  are  near  enough 

to  be  marked.     But   the 

student    must    fully   un- 
derstand that  a  point  too 

far  away  to   be   marked 

can   be  mentally  located, 

and  the   lines   drawn 

toward    it    with    closely 

approximated  accuracy. 

The  essential  thing  is  to 

have  the  position  of  such  a 

point  dearly  thought  out, 

—  even,  for   instance,  as 

specifically  as   that  it  is 

"  the  width  of  the  board," 

or   "  three    times "    that, 

distant.      The    power   to 

do  this  accurately  grows 

rapidly,     and     can     be 

attained      by    students 

of  moderate   ability.     It 

is    one    object    of    this 

study. 

Rendering  from  the  Print.  —  As   more   complex    sketches   are 

made,  certain  parts  may  be  expressed  in  color  (that  is,  covered 

with  a  tone  of  pencil  lines),  as  was 
done  with  the  title  space  of  the 
books  in  Chapter  XIX.  The  door- 
way, windows,  and  shaded  sides  of 
the  buildings  in  this  exercise  (Fig. 
131),  are  examples  of  this.     Such 

use  of  color  is  intended  sometimes  to  attract  the  eye  to  the  most 

important  or  interesting  parts,  or  to  bring  out  the  beauty  of  such 

details  as  the  majestic  forms  of  the  trees  in  A,  Fig.  132. 

83 


Fig.  132 


FREEHAND    PERSPECTIVE 

The  Comparative  Simplicity  of  Perspective.  —  By  experience  in 
mentally  grouping  each  new  vanishing  line  with  the  set  to  which 
it  belongs,  the  perspective  of  apparently  difficult  studies  becomes 
simple.  In  Fig.  133  a  seemingly  complex  group  of  buildings  is 
shown  to  need  but  four  vanishing  points  for  nearly  all  of  its 
lines. 


84 


Chapter  XXIV 

TYPE    FORMS    HELPFUL   IN   UNDERSTAND- 
ING THE  HOUSE  ^  — THE  SQUARE  FRAME 


T 


HE  Model.  —  The  model  for  the  square  frame  is  six  inches 
on  a  side,  and  one  inch  square  in  section.  Looked  at 
from  the  front, 


it  appears  as  two  con- 
centric squares  one 
inch  apart  (Fig.  135). 
It  is  placed  with  one 
set  of  long  edges  ver- 
tical, and  the  other 
horizontal  and  mak- 
ing angles  of  sixty 
degrees  to  the  left 
with  the  picture  plane 
(Fig.  136). 

In  considering  its 
shape  it  may  be  first 
regarded  as  a  Plinth, 
or  one-inch  rectangu- 
lar slice  from  a  six- 
inch  cube,  and  there- 
fore one  sixth  of  the 
cube  in  thickness 
(Fig.  137). 

^  Some  of  the  geometric 
solids  here  and  later  given  may 
be  omitted  at  the  discretion  of 
the  teacher.  Those  selected 
for  study  should  be  such  as  to 
supply  any  deficiencies  in  the 
student's  mastery  of  the  subject. 


iaia»soKi^^Viz:xsSiS¥i^»^¥iSs:^->^S:^:S:S»ij,a:iVSi^Ui, 


Fig.  134 


85 


FREEHAND    PERSPECTIVE 


A           ,/ 

z 

Y 

B 

TRONT 

Fig.  135 


Fig.  13G 


Drawing  the  Model.  —  The  lines  of  this  solid  may  be  drawn 
and  its  proportions  established  in  the  same  manner  as  those  of 
the  cube  (p.  54).  Remember  to  place  the  eye 
level  at  once  after  drawing  the  first  vertical  edge. 
Like  the  cube,  these  type  forms  should  be  lightly 
sketched  first  and  later  rendered  with  the  firm, 
simple  lines  appropriate  to  them.  Being  more 
complicated  than  the  cube,  their  visible  edges 
may,  if  necessary,  be  strengthened  (that  is,  be  drawn  as  they 
would  finally  appear)  as  soon  as  determined,  to  avoid  confusion. 

The  inner  square  of  the  frame  may  ne:!rt  be  marked 
out  on  the  plinth  (Fig.  137).  To  do  this  place  points 
one  sixth  of  the  front  vertical  edge  from  each  end, 
and  from  them  vanish  lines  to  VPl.  In  these  lines 
the  edges  A  and  B  of  the  inner  square  must  lie.  By 
the  front  view  (Fig.  135),  we  perceive  that  the  corners 
of  the  inner  square  lie  in  the  diagonals  of  the  outer 

one.  One  diagonal,  C,  will  mark  two  corners 
{x  and  y)  of  the  inner  square.  Its  other  two 
corners  are  found  by  drawing  the  vertical  edges 
of  the  inner  square,  from  corner  x  down  to  line 
B  and  from  y  up  to  A. 

If  this  inner  square  were  cut  out,  leaving  a 
frame,  parts  of  the  inner  thickness  of  the  frame  could  then 
be  seen.  Of  this  inner  thickness,  line  D  (Fig.  138)  lies  in  the 
back  surface  of  the  frame,  parallel 
to  B,  and  at  actually  the  same 
height.  Hence  it  can  be  started 
at  a  point  [z)  on  the  right  hand 
vertical  edge  of  the  frame,  obtained 
by  drawing  line  E  from  the  near 
end  of  B  to  VP2.  This  may  be 
called  carrying  line  B  "  around 
the  corner."  From  this  point  z^  D  is  drawn  to  YPl.  The 
lower  inner  edge  (F)  at  the  back  is  parallel  to  the  outer  thick- 

86 


Fig.  137 


Fig.  138 


TYPE    FORMS    HELPFUL,    ETC 


ness  edges,  and  therefore  can  be  drawn  to  VP2,  cutting  off  line 
D.  A  vertical  line  from  where  D  and  F  meet  completes  the 
inner  thickness. 

Tests.  —  The  correctness  of  this  drawing  can  be  tested  by 
adding  the  invisible  portions,  shown 
by  dotted  lines  in  Fig.  138.  Thus 
if  line  A  be  carried  around  the  cor- 
ner, giving  the  invisible  edge  H,  the 
inner  invisible  edge  I  should  cut  it  in 
line  Gr  continued. 

The  Application  of  Type  Form 
Principles.  —  The  application  of  the 
foregoing  work  to  the  drawing  of 
such  parts  of  the  house  as  windows  and  doors  may  be  seen 
in  Fig.  139,  where  the  inner  edges  of  door  and  window  frames 
converge  with  the  set  of  horizontal  lines  at  right  angles  to  the 
door.  For  instance,  lines  A  and  B  converge  with  the  dormer 
eaves  and  other  lines  tending  to  VP2,  and  line  C  vanishes  with 
the  set  to  VPl. 


Fig.  139 


87 


Chapter  XXV 


THE  SQUARE  PYRAMID  AND  SQUARE 

PLINTH 


T 


HE  Models.  —  The  plinth  is  two  inches  high  and  six  inches 
square :  the  pyramid  four  inches  square  at  base  and  eight 
inches  high.     The  models   can  be   made  (Fig.  141).    If 

made,  note  that  in 
order  to  secure  the 
required  height  in 
the  completed  pyra- 
mid, the  length  [xij 
in  Fig.  141)  of  each 
triangular  side  piece 
of  the  pyramid  pat- 
tern is  measured 
from  xy  in  Fig.  142, 
where  the  true 
length  of  a  side  face 
is  shown.  The  face 
x-y-3  in  Fig.  142 
leans  back,  making 
x-o  foreshortened. 

Position.  —  The 
plinth  rests  on  one 
square  face,  with  its 
sides  at  angles  of 
thirty  and  sixty 
degrees  with  the 
picture  plane  (Fig. 
,.!  142).  The  pyramid 
Fig.  140  stands  on  the  plinth, 

88 


SQUARE    PYRAMID    AND    SQUARE    PLINTH 


T 
i 


A.  Pattern 

FOR  MAKING 
PYRAMID 


«\rith  its  base  parallel  to  and  equidistant  from  the  edges  of  the 
plinth  top. 

Drawing  the  Models.  —  Proceed 
with  the  plinth  as  with  the  cube, 
remembering  that  its  height  is 

but  one  third  as 

much  as  the  cube 

in  proportion  to 

its  breadth. 
For  the  pyra- 

mid  base,  one 

perspective  sixth 

must  be  marked 

from   each   end 

on  line  AB  (Fig. 

143).     To    get 

these  points   (1  and  2)   a  diagonal  of  the   side 

ABCD  can  be  used  (as  in  Ch.  XX).    The  vertical 

edge,  AC,  being  unforeshortened,  is  first  divided 

into  six  actually  equal  parts.     Lines  from  the  up- 


f- 


B.  Pattern  for 

MAKINO   PLINTH 


16  in. 


Fig.  141 


B.   [Plan 

Fig.  149 


EVg     LIVCl- 


ii£7 


per  and  lower  division  points 

(8  and  9)  to  VPl  transfer  these 

divisions  proportionately    to 

the  diagonal,  AD.     From  the 

diagonal  they  are  transferred 

by  vertical  lines  to  the  edge 

AB.     (See  also  Fig.  144.) 
Or  a  diameter  through  the 
center  (O) 

will  divide  f^^-  i*^ 

AB   into  perspective  halves  at  3,  when  each 
half  can  be  divided  into   thirds  by   the   eye. 
One  method  can  loe  used  to  prove  the  other. 
From  points  1  and  2  draw  lines  to  VP2.     Where  they  cut  a 

diagonal,  as  AE,  will  be  two  corners  (4  and  5)  of  the  pyramid. 

89 


Geometric  view  or 

SIDE  piF  PLINTH  . 
Fig.  144 


FREEHAND    PERSPECTIVE 


Lines  to  VPl  through  these  points  will  give  the  two  other  corners 

(6  and  7),  and  complete  the  base  of  the  pyramid. 

The  apex  of  the  pyramid  will  be  vertically  over  the  center  of 

its  base,  point  O.     On  a  vertical  line  from  O  must  be  measured 

the  perspective  height  of  the  pyramid.  Its 
actual  height  is  four  times  that  of  the  plinth. 
The  nearest  corner  of  the  plinth  is  convenient 
to  use,  hence  from  A  the  vertical  line  is  con- 
tinued, and  four  times  AC  is  measured  on  it, 
giving  AG  for  the  pyramid  height  as  it  would 
appear  at  that  point.  If  now  this  height,  AGr, 
could  be  moved  back  on  the  diagonal  AE  to  O 
it  would  appear  to  shorten  as  moved.  Its  top 
would  describe  an  actually  horizontal  line 
above  AE,  that  is,  a  line  parallel  to  it,  conse- 
quently vanishing  to  the  same  point.  There- 
fore continue  the  diagonal  AE  to  the  eye  level, 
giving  VP3,  and  draw^  the  parallel  line  from 
G  to  VPS,  which  will  mark  on  the  vertical 
from  0  the  desired  perspective  height  at  x. 
Complete  the  pyramid  by  drawing  its  oblique 
edges  to  the  corners  of  its  base. 

Applications.  —  The  difficulty   of   making 


Fig.  145 


a  church  spire  or  a  tower  (Fig. 
145)  "  stand  true  "  will  be  readily 
recognized.  The  use  of  the  di- 
agonals (AB  and  CD)  will  aid 
in  placing  its  axis  and  apex. 

Any  upright  rectangular  ob- 
ject with  equally  sloping  sides  is 
easily  constructed  on  the  same 
principle,  as  the  grape  basket  in 
Fig.  145a.  If  such  an  object  is 
leaning,  the  meeting  of  its  cor- 
ners continued  can  still  be  used 
as  there  shown. 


6PAPr-BA«K»T 


Fig.  143a 


90 


Chapter   XXVI 


THE    SQUARE    FRAME    LEANING   ON  THE 

RECTANGULAR  BLOCK  — A  PROBLEM 

FOR  ORIGINAL  STUDY 

THE  Models.  —  These  have  ah^eady  been  described  on  pages 
48  and  85,  respectively. 
Position —  The  block  rests  on  one  long  face,  with  its 
square  ends  making  angles  of  sixty  degrees  with  the  picture  plane. 


Plak 


Fig.  146 


The  frame  leans  against  the  block,  equidistant  from  its  ends, 
and  with  a  distance  equal  to  half  the  width  of  the  block  between 
its  lower  edge  and  the  block. 


91 


Chapter  XXFII 


Fig.  1-17 


CYLINDRICAL  OBJECTS  WHEN  NOT 
VERTICAL 

A  LTHOUGrH  in  Chapter  III  the  cylinder  held  horizontally 
/%      was  mentioned,  we  have  only  studied  cylindrical  objects 

^    JL   when  vertical.     In  this  position  they  have  been  found 

symmetrical,   the   ellipses  and  the   axis   (which   is  the  middle 

from  end  to  end)  being  at  right  angles  to  each  other.     To  study 

them  in  other  positions  begin  with  the 
cylinder  model  held  horizontally,  with  its 
middle  on  the  eye  level,  and  its  ends 
equally  distant  (Fig.  147).  It  will  be  read- 
ily seen  that  the  ends  now  appear  at  right 
angles  to  the  axis,  as  when  the  object  was 
vertical.     Turning  it  a  little  so  the  right 

end  can  be  seen  (still  keeping  it  horizontal  and  at  the  eye  level), 

it  will  be  observed  that  the  apparent  directions  of  the  axis  and 

ellipses  are  unchanged.     (The  axis  being 

a  horizontal  line   and   at  the  eye  level 

remains   apparently   horizontal,  and  the 

ellipses  still  appear  vertical.     The  further 

ellipse  has  become  a  little  shorter  and 

rounder,  and  the  side  boundaries,  like  all 

parallel  receding  horizontal  lines,  appear 

to  converge  to  the  eye  level.) 

Now  lower  the  model,  keeping  it  turned  away,  till  it  rests  a 

foot  below  the  eye  on  some  horizontal  support  (as  the  box  in 

Fig.  149). 

The  side  boundaries  and  the  axis,  being  below  the  eye,  vanish 

upward  to  a  point  on  the  level  of  the  eye.     They  will  continue 

92 


Fig.  148 


CYLINDRICAL    OBJECTS    NOT    VERTICAi. 


If  WJ  I  I- 

UMmMliliiuif 


to  vanish  to  the  eye  level,  whether  below  or  above  the  eye  (B 

in  Fig.  149),  as  long  as  the  cylinder  is  kept  horizontal.     The 

ellipses   should  now  be  exam- 
ined to  see  if  in  this  position 

they  appear,  as  formerly,  to  be 

at  right  angles  to  the  axis.     Do 

this  first  with  the   head   erect 

as    usual,    looking    with    care, 

and   deciding   mentally.     Then 

try  inclining  the  head  (in  this 

case  of  A,  Fig.  149,  to  the  right 

and   downward)    to    bring   the 

face   in   relation   to  the  model 

as  it  would    be   if    both  were 

vertical.     Two  pencils  held  in 

the  shape  of  a  letter  T,  held  in 

front  of  the  cylinder  (as  in  Fig. 

150) 
will 
help 

make  sure  that  the  axis  of  the  cylin- 
der and  the  long  diameters  of  its 
ellipses  unmistakably  appear  at  right 
angles  to  each  other. 

To  understand  how  this  can   be 
the    case, 
hold    the 

cylinder   again   at  the  eye  level,  as 

in  Fig.  151,  and  with  a  pencil  mark 

on  it  the  points   (A  and  B)   where 

the  side  boundaries   meet  the  ends 

of  the  ellipse.     Lower  the  cylinder 

again  (Fig.  152)  when  it  will  be  seen  that  these  points  are  not 

now  at  the  ends  of  the  ellipse,  and  that  the  line  AB  is  not  now 

its  long  diameter.    It  has  now  a  new  long  diameter,  CD,  at  right 

93 


.iiiMim 


LUJlU 


m- 


Fig.  149 


Fig.  151 


FREEHAND    PERSPECTIVE 


cTBKps. 


angles  to  the  axis  in  its  new  apparent  direction;  and  also  new 
side  boundary  lines  from  C  and  D  toward  the  vanishing  point. 

As  the  long  diameter  is 
always  at  right  angles  to  the 
axis,  it  must  change  its  position 
when  the  object  is  moved  so  that 
its  axis  appears  changed  in 
direction.  We  see  therefore 
that  tlie  long  diameter  is  movable. 
These  experiments  may  be 
tried  with  other  cylindrical  ob- 
jects, as  a  tumbler,  or  the  flower  pots  in  the  next  chapter.  The 
leaning  dish  in  Chapter 
YIII,  and  the  tilted  cover 
in  Chapter  X  are  examples. 
It  will  invariably  be  found 
that,  provided  the  circular  de- 
tails   of  a 


Fig.  152 


Fig.  1o4 


cylindrical  ^^^-  ^^^ 

object  are  actually  at  right  angles  to  its  axis,  they 

ivill  appear  so  whatever  the  position  of  the  object. 

Consequently,  cylindrical  objects  alivays  appear 

symmetrical. 

Tests.  —  A  drawing  of  such  an  object  may 
be  tested  by  turning  it  to  bring  the  object  in 
question  vertical  (Fig.  156,  Ch.  XXVIII),  when 
errors  in  symmetry  will  be  more  apparent. 

A  wheeled  vehicle  (Fig.  153)  is  a  common 
illustration  of  this  principle;  also  a  clock  (Ch. 
XXXI)  and  the  round  arches  in  Chapter  XXXII. 
Others  will  readily  occur  to  the  student. 
Flowers  (Fig.  154)  are  striking  examples,  and 
many  awkward  drawings  of  flowers  are  so  be- 
cause drawn  in  ignorance  of  this  beautiful  and 
simple  principle  of  the  symmetry  of  the  cylinder. 

94 


Chapter  XXVIII 
A  GROUP  OF  FLOWER  POTS 

S  with  previous  examples,  drawing  this  exercise  is  optional 
with  the  student,  according  to  his  proficiency.  But  he 
should  compose  a  similar  group,  that  is,  having  in  it  at 


Fig.  155 

least  one  cylindrical  object  not  vertical. 
And  he  should  take  especial  pains  to  se- 
cure  the   symme- 
try of  such   non- 
vertical  objects. 

The  illustra- 
tion (Fig.  156) 
shows  how  this 
symmetry  may  be 
tested  by  turning  Fig.  ise 

the  group. 

95 


A. 


Chapter  XXIX 


THE  CIRCULAR  FRAME  IN  A  SQUARE 

FRAME 


T 


HIS  is  an  example  of  rectangular  and  cylindrical  forms 
in  the  same  object.  The  explanation  should  be  carefully- 
studied,  and  the   exercise  drawn  unless  the  student  is 

^    experienced. 

The  Circular  Frame.  — 
.     After  the  square  frame 
I     (Ch.   XXIV)   is    drawn, 
I    look  at  the  model  from 
f    the  front  (Fig.  158),  and 
\    note  that  the  outer  sur- 
f    face  of  the  ring  touches 
I    the  square  frame  at  four 
I    points  only  —  where  the 
i    diameters  of  the  square 
5    cross  it  (A,  B,  C,  and  D). 
i    These    diameters,   being 
\    parallel    respectively   to 
\    the  sides  of  the  square, 
\    are  represented   in    the 
I    perspective  by  a  vertical 
t    line,  and  a  line  vanish- 
j    ing  in   VPl,  both  pass- 
j    ing  through  the  true 
i    center  O,  at  crossing  of 
^    the  diagonals.     Through 
these    four    points    the 
outer    edge   of   the    cir- 
cular frame  must  pass. 

96 


Fig.  1.57 


CIRCULAR  FRAME  IN  A  SQUARE  FRAME 


1   - 

A 

-c 

^G 

s^ 

8- 

V^ 

J, 

1 

Disregarding  at  first  the  opening  in  it,  the  circular  frame  may- 
be temporarily  thought  of  as  a  slice  from  the  cylinder,  one  inch 
thick,  and  therefore  a  very  short  cylinder,  with 
an  axis  only  one  inch  long.  As  placed  within 
the  square  frame  (Figs.  157  and  159)  its  axis  and 
side  boundaries  are  parallel  with  the  short  edges 
of  the  square,  and  its  actual  centre  and  that  of 
_  _._..  the  latter  coincide. 

Hence  we  may  draw  its 
axis  from  this  center  (0) 
to  VP2.  Its  circular  outer  edge  will 
be  seen  as  an  ellipse  at  right  angles  to 
this  axis,  and  passing  through  the  four 
points  A,  B,  C,  and  D,  previously  found. 
As  the  short  diameters  of  such  ellipses 
always  appear  to  lie  in  a  line  with  the 
axis  of 
the 


Front  view. 
Fig.  158 


Fig.  159 


axis 


,.,.^ 


Top  AS  SEEN  SY  EYE, 

lookino  at  a . 
Fig.  160 


the  object  (Fig.  160), 
line  ivill  give  the  apparent  direc- 
tion of  the  short  diameter  of  the 
ellipse.  (Here  the  beginner  is 
advised  to  turn  the  paper  round, 
bringing  the  axis  as  a  vertical 
line,  the  better  to  secure  the 
symmetry  of  the  cylindrical  part 

of  the  model.     See  Fig.  161.) 

To  obtain  the  length  of  this  short 
diameter,  slightly  curved  lines,  perpen- 
dicular to  the  short  diameter,  are 
sketched  from  B  to  the  right  and  from 
C  to  the  left,  giving  x  and  ij  for  the 
extreme  front  and  back  of  the  ellipse. 
Now  (since  an  ellipse  is  always  symmet- 
rical) the  longest  dimension  or  long  diameter  of  this  ellipse  is  not 
on  the  vertical  line  through  A  and  D  (Fig.  159),  but  is  on  a  line  at 


Fig.  161 


97 


S 


FREEHAND    PERSPECTIVE 


right  angles  to  x  ?/,  and  through  its  apparent  middle  from  front 
to  hack;  making  one  end  fall  in  front  of  D,  and  the  other  back 
of  A.  It  will  also  be  a  little  in  front  of  O,  the  true  center  of 
the  circle,  and  a  little  longer  than  A  D,  but  not  touching  the 
frame.  Mark  lightly  and  accurately  this  apparent  middle  and 
sketch  the  long  diameter  through  it.  Mark  the  ellipse  ends 
by  sketching  rounded  curves,  from  A  back  and  from  D  for- 
ward, making  them  symmetrical  on  the  long  diameter,  and 
equidistant  from  the  axis.  Complete  the  ellipse  by  connecting 
these  ends  and  sides,  correcting  if  necessary,  till  the  ellipse  is 
perfect. 

For  the  inner  ellipse  proceed  as  with  the  top  of  the  cylinder 
in  Chapter  IV,  remembering  to  make  its  proportions  as  shown 
in  Fig.  158 ;  also  that  the  perspective  halves  of  the  short  diameter 
are  already  found  by  the  true  center,  O.  Or  four  points  for  the 
inner  circle  (e,  e',/and/.  Fig.  161a)  may  be  found  by  the  method 
given  for  concentric  circles  in  Chapter  XX.  Thus  points  E  and 
F  are  set  off  their  proportionate  height  on  the  nearest  vertical 
edge  xy,  and  lines  vanished  from  them  to  VPl.  Where  these  lines 
cross  AD  are  e  and  /,  the  two  points  for  the  inner  circle  corre- 
sponding to  A  and  D  in  the  outer  circle.  For  the  points  on  its 
horizontal  diameter  (e'  and  /',  corresponding  to  C  and  B  in 
the  outer  circle)  vertical  lines  are 
drawn  from  the  points  (1  and  2) 
where  the  vanishing  lines  cross  the 
diagonal. 

The  inner  ellipse  is  then  con- 
structed as  was  the  outer  one.  ^j^  Fig.  leia 
'  The  side  boundary  lines  of  the  circular  frame  may  now  be 
drawn  to  VP2,  when  there  will  remain  only  its  inner  thickness  to 
draw.  The  edge  of  this  is  an  inner  circle  on  the  back  of  the 
frame,  actually  like  the  inner  one  in  front.  Draw  lines  (E  and  F) 
from  the  ends  of  the  front  inner  ellipse  to  VP2.  These  may  be 
called  side  boundary  lines  of  the  cylindrical  opening ;  and  their 
convergence  measures  the  smaller  length  of  the  desired  back  el- 
lipse, which  lies  with  its  ends  in  these  lines  as  does  the  front  one. 

98 


K 

/ 

c 

fc 

X' 

/>« 

S 

^  k 

J 

^' 

K. 

'y 

/ 

b 

CIRCULAR  FRAME  IN  A  SQUARE  FRAME 

The  round  arch  (Fig.  162,  also  Ch.  XXXII)  is  an  interesting 
application  of  this  principle.  Errors  in  drawing  these  and 
kindred  forms  (B  in  Fig.  162)  so  common  with  beginners  are 
easily  avoided  when  these  principles  are  understood. 


^°  J- 


A-  Showing  application  of  methods. 


B.  WRONO.  INNER  BACK  ELUPSE 
MADt  Too  5H0RT,  AS  IF  ACTUALW 
SMALLER,  INSTEAD  OF  A  DUPLICATE. 


Fig.  162 


99 


T 


Chapter  XXX  . 
A    ROUND   WINDOW 

HIS  exercise  (Fig.  163)  and  the  methods  of  sketching  it 
should  be  carefully  studied,  and  if  necessary  it  should 
be    drawn.      After    this   the    student    should    sketch    a 

similar      example 
k  from  a   building  or 

photograph. 

Having   drawn 
^''  '  the  straight-line  part 

i    i  >  of  the  exercise,  the 

round  window  is 
'^--        "  (  ^\  \\^  next    to  be   consid- 

^  ered.     This  is  actu- 

ally a  cylindrical 
<^P''  •  ^  opening  in  the  wall. 

Being  above  the  eye, 
Affc..,  .''^M^''"  the  circles  of  thewin- 


~^ 


dow  appear  as  slant- 
ing ellipses  like  the 
cylinder  ends  in  Fig. 
.^i,  149  (Ch.  XXVII). 

^''^  ^  To  make  sure  that 

the    slant    of    these 
ellipses   shall   agree 
with  the  straight-line 
[  part  of  the  building, 

U  the  axis  of  this  cylin- 

,j      X  drical  window  is 

''  ^'-h  used.     This    axis  is 

S^Ji  actually  at  right  an- 

^^^-  ^^^  gles  to  the- wall,  and 

100 


A    ROUND    WINDOW 


is  therefore   parallel    to   the   lines   already  vanishing   in  VPl. 

Hence  the  apparent  middle  of   the  window  ellipse  (O   in   Fig. 

164)  is  first  located,  and  the  axis  is  i 

drawn  through  it  to  VPl,  extend- 
ing forward  indefinitely.     The  long 

diameter  (AB)  is  sketched  at  right 

angles  to   the  axis,  and  the  short 

diameter   (CD)    is  set  off   on  the 

axis  line.    The  ellipse  is  then  drawn 

through  these   four    points.      The 

inner  ellipses  of  the  window  are 

shorter  as 
well  as  far- 
ther back 
than  the 
outer  one. 

For  the 
partial  ellip- 
ses   of    the 

quatrefoil,  the  actual  center  of  the  inner 
ellipse  (1  in  Fig.  165)  must  be  marked, 
and  through  it  a  vertical  line  (EF)  and  a 
horizontal  one  (GH,  vanishing  in  VP2) 
drawn.  The  real  centers  of  the  quatrefoil 
circles  fall  each  on  one  or  the  other  of 
these  lines.  Their  long  diameters  (a  little 
in  front  of  these  real  centers)  are  parallel 

with  the  other  long  diameters  of  the  window. 

This   quatrefoil   is   especially  an   example   of  objects  which 

should  in  practice  be  sketched  freehand  first,  and  afterward  tested 

by  the  constructive  methods  here  given.     (See  Introduction.) 


A.  FRONT  VIEW 


Fig.  164 


'-^^ 


PERSPecTive 
Fig.  165 


101 


T 


f*^ 


T" 


) 

7 

f 
\ 

M 

1 

n 

\ 

1 

3 

, 

% 

1 

11 

j 

t 

s 

;i 

i 

f 

Chapter  XXXI 
THE  CLOCK  A  PROBLEM 

HIS  example  (Fig.  166)  is  given  as  an  aid  in  rendering, 

though  it  may  be  drawn  first  if  desired. 

The  Model.  —  Any  clock  containing  rectangular  forms 

—  '     -    ""    '■■■--    and  the   usual   cir- 

cular face  will  serve 
as  the  model.  It 
should  be  placed 
above  the  eye. 

Conditions. — Un- 
like proJolems  in 
general,  this  draw- 
ing may  be  made 
from  the  object, 
but  must  be  done 
without  assistance. 
The  aim  is  to  test 
the  student's  abil- 
ity to  apply  the 
principles  taught 
in  the  last  few 
chapters. 


Fig.  166 


102 


Chapter  XXXII 
THE  ARCH 

THESE  arches  from  the  cloister  of  St.  Paul's  Without 
Gates,   at  Rome,   also    illustrate   the   symmetry  of 
cylinder,   and  can    be  drawn    by  the  same  method 


the 

the 

as 


Fig.  167 


the  round  window  in  the  last  chapter.  (See  Ch.  XXX.)  They 
are  semi-cylinders  and  their  openings  are  semicircles  (Fig.  169). 
The  semicircles  are  sketched  on  a  horizontal  line  (A)  which, 

103 


FREEHAND    PERSPECTIVE 

being  above  the  head  in  this  instance,  vanishes  downward  (Fig. 
168).  The  true  centers  of  the  semicircles  are  on  this  line,  and  from 
these  centers  the  axes  are 
drawn,  vanishing  with 
other  lines  to  VPl.  The 
joints  of  the  stones  form- 
ing each  arch,  being  lines 
really  tending  to  meet  in 
its  true  center,  are  so 
drawn  in  perspective. 

Pointed  arches,  and 
other  modified  forms  can 
be  readily  drawn  on  the 
same  principles  here 
used. 

The  student  should 
draw  this  exercise  unless  experienced,  when  he  should  instead 
select  a  print  of  artistic  interest,  illustrating  the  same  principle, 
and  make  from  it  a  careful  and  expressive  sketch.  In  either 
case,  he  should  follow  his  first  drawing  with  another  involving 
the  use  of  this  principle,  from  a  building. 


Fig.  168 


Front       Viemv 
Fig.  169 


104 


Chapter  XXXIII 


INTERIORS 


A  ROOM  PARALLEL  TO  THE 
PICTURE  PLANE 


THE  Cube  as  a  Model.  —  With  a  penknife  loosen  one  face  of 
the  cardboard  cube,  and  turn  it  back  or  take  it  off.  Place 
it  within  a  foot  of  the  eye,  with  the  opening  parallel  to 


the  picture  plane,  and  the  eye  level  a  little  less  than  two  thirds 
of  the  way  up.  If  desired,  the  v^indows,  doors,  rug,  and  pictures 
may  be  marked  with  a  pencil  on  the  inside  of  the  cube.  It  now 
serves  to  illustrate  the  room  shown  in  Figs.  170  and  172,  as  the 
cubes  and  prism  illustrated  a  house  in  Chapter  XXII.  In  this 
room  the  floor,  ceiling,  side  walls,  and  all  details  on  their  surfaces 

105 


FREEHAND    PERSPECTIVE 


(as  the  window,  the  side  door,  and  the  rug)  are  foreshortened; 
while  the  side  edges  converge  to  VPl  directly  in  front.  The  back 
wall  and  all  surfaces  parallel  to  it  (as  the  end  of  the  table  and 
one  side  of  the  stool),  being  parallel  to  the  picture  plane,  appear 
in  their  true  shape. 

Directions.  —  This  example  (Fig.  170)  should  be  drawn  by  the 
student  if  a  beginner.     After  this  the  end  of  a  room,  also  with  its 
farther  wall  parallel  to  the  picture  plane,  should  be  drawn  from 
memory   or    invention.^      A   hall,    a 
kitchen,  a  street  car,  or  a  piazza  w^ill 
be  recognized  as  especially  adapted 
to  such  views. 

The  Apparent  Width  of  the  Sides. 
—  To  aid  in  estimating  this,  recall  the 
appearance  of  the  most  foreshortened 
side  of  the  cube  in  Chapter  XVI. 
This  estimate  may  then  be  tested 
by  pencil  measurement  of  the  cube 
model. 

The  Pictures.  —  Although  the  pic- 
ture on  the  right  wall  is  inclined 
slightly  forward,  its  sides  are  still  parallel  to  the  picture  plane ; 
and  therefore  appear  in  their  true  direction  and  shape.  With 
the  picture  on  the  back  it  is  not  so.  Its  top  is  slightly  nearer  to 
us  than  its  lower  edge ;  and  must  therefore  appear  longer,  mak- 
ing the  sides  appear  to  converge  downward  (though  almost 
imperceptibly) .  To  determine  the  direction  of  this  convergence, 
revolve  it  forward  on  its  lower  edge  in  imagination  until  hori- 
zontal (Fig.  171).  It  will  at  once^,be  seen  that  the  sides  (A  and 
B)  in  this  position  are  parallel  with  the  lines  already  vanishing  in 

^  It  may  be  asked  why  memory  oi-  inventive  drawings  should  be  advised  before  study  from 
a  room.  But  in  this  case  we  meet  a  subject  of  which  all  have  something  in  memory;  and 
drawing  from  memory  when  possible  (when  one  has  something  remembered)  is  not  only  far 
pleasanter,  but  much  less  laborious.  The  object  or  place  itself  presents  to  a  beginner  a  confus- 
ing mass  of  detail,  much  of  it  not  needed  for  the  drawing.  The  student  must  make  later  many 
drawings  from  the  place  to  accumulate  knowledge  ;  but  will  always  do  his  most  free  and 
individual  work  from  this  knowledge,  not  directly  from  the  object. 

106 


Fig.  171 


ROOM    PARALLEL    TO    PICTURE    PLANE 


VPl.  If  now  the  picture  should  be  slowly  revolved  upward,  the 
converging  point  for  the  sides  would  descend  as  the  picture  rises. 
As  the  picture  is  not  moved  sidewise  at  all,  this  point  of  con- 
vergence (OVPl)  can  only  move  downward  in  a  vertical  line 
from  VPl  like,  the  roof  ends  in  Chapter  XXII.  When  the 
picture  is  returned  to  its  original  position,  it  varies  but  slightly 
from  the  vertical ;  consequently  OYPl  is  too  far  away  to  locate 
and  the  vanishing  of  its 
sides  must  be  estimated. 
Make  sure  it  is  slight 
enough,  and  toward  a 
point  vertically  under 
VPl. 

An  Open  Door.  —  So 
far  we  have  found  but 
one  vanishing  point  on 
the  eye  level.  If  we 
begin  to  open  the  far- 
ther door,  its  horizontal 
lines  will  instantly  ac- 
quire a  vanishing  point, 
but  at  an  infinite  dis- 
tance. In  proportion  as  the  door  swings  toward  being  parallel 
with  the  sides  of  the  room  (that  is,  with  the  direction  of  seeing) , 
this  vanishing  point  will  move  inward.  When  the  door  becomes 
quite  parallel  with  the  walls  of  the  room,  this  point  will  coincide 
with  VPl.  If  the  door  is  swung  still  farther  back,  this  point 
moves  on  toward  the  left.  The  apparent  width  of  the  door 
thus  opened  may  be  measured  by  an  ellipse  on  the  floor 
(Fig.  172)  representing  the  path  of  its  near  corner  as  it  swings 
in  a  circle.  The  short  diameter  of  this  ellipse  is  proportioned 
to  the  foreshortening  of  the  floor  in  which  it  lies.  It  can 
also  be  found  by  the  use  of  lines  parallel  to  the  floor  diagonal 
(Fig.  173).  From  this  diagram  it  is  seen  that  since  the  floor  is  a 
square,  its  diagonal  is  a  line  at  45°  with  its  sides.     By  lines  at  45°, 

107 


Fig.  m 


FREEHAND     PERSPECTIA^E 


distances  into  the  picture  can  be  measured  equal  to  others  paral- 
lel to  the  picture  plane  (Fig.  103,  p.  65).  In  Fig.  173  (and  per- 
spectively  in  Fig.  172)  AC  is  made  equal  to  AB  by  drawing  BC  at 

c  45°,  or  parallel  to  the  floor  diagonal.^ 

I  /"'^  I    ^^>/^|  It  will  also  be  noted  that  the 

thickness  edges  of  the  door,  being 
horizontal  and  at  right  angles  to 
its  top  and  bottom,  have  their 
own  vanishing  point  upon  the  eye 
level  (VP4).  Also  the  door  knob 
is  cylindrical,  and  its  axis  is  paral- 


~SSZSZZZZZZZZZZZZZ2ZZZZZA 

A 


iM—-\' 


\^J^ 


,0^ 


6i 


n 


lei  to  these  edges. 


Fio.  173 


The  Stool.  —  The  proportion  of 
the  stool  is  found  in  the  same 
manner  as  the  height  of  the  pyra- 
mid in  Chapter  XXV.  The  room 
may  be  considered  as  nine  feet  high,  and  the  stool  as  approxi- 
mately eighteen  inches,  or  one  sixth  the  height  of  the  room. 
Mark  any  point  (as  H  in  Fig.  172)  where  it  is  desired  to  place  it 
on  the  floor  of  the  room.  Its  height  cannot  be  compared  directly 
with  that  of  the  room  here,  for  we  cannot  determine  where  a 
vertical  line  from  H  will  touch  the  ceiling.  Therefore  imagine 
the  stool  moved  from  point  H  in  a  straight  line  to  any  place  on 
the  front  edge  of  the  room,  as  I,  where  its  height  (IJ)  can  be 
measured  by  that  of  the  room.  If  now  it  were  moved  back  on  this 
same  line  (IH) ,  its  top  would  move  in  a  horizontal  line  directly 
over  IH,  that  is,  actually  parallel  to  it,  or  vanishing  in  the  same 
point  on  the  eye  level.  Both  lines  (HI  and  one  from  J)  may 
therefore  be  carried  to  this  point  VP5.  The  height  of  the  stool, 
when  placed  at  any  point  on  the  line  from  1  to  VP5  will  be 
the  vertical  distance  (as  at  H)  between  these  lines. 

The  near  side  of  this  stool  is  now  drawn  in  its  true  shape,  and 
the  parts  at  right  angles  to  this  side  found  by  vanishing  lines 
to  VPl. 

*  The  ellipse  is  made  horizontal,  as  explained  in  Chapter  XLIII. 

108  ,  ^ 


ROOM    PARALLEL    TO    PICTURE    PLANE 

Finally,  to  improve  the  composition  some  of  the  ceiling  and 
a  little  of  the  floor  are  cut  oif ,  as  shown  by  the  dotted  lines  in 
Fig.  172.  This  gives  a  more  generally  favorable  shape  (Fig.  170) 
to  the  inclosnre,  and  keeps  the  oblique  lines  from  running  to  its 
corners,  which  should  always  be  avoided.     (See  p.  116.)^ 

*  See  Chapter  XLII  for  fiu-ther  coDsideratiou  of  a  room  parallel  with  the  picture  plane. 


109 


Chapter  XXXIV 


INTERIORS  CONTINUED  — A  ROOM  AT 
ANGLES  TO  THE  PICTURE  PLANE 

THE  Model.  —  The  cube  model  may  be  prepared  for  illus- 
trating this  study  by  removing  a  side  adjacent  to  the 
opening  made  for  the  previous  study. 

Position.  —  The 
■j::=^e-'-  v-'^"  ^^-^  -s     eye  level  and  the  dis- 

tance from  the  eye 
are  the  same  as  in 
the  last  chapter. 
Place  the  model  so 
that  its  receding 
faces  make  very  un- 
equal angles  with  the 
picture  plane  (B  in 
Fig.  175).  Both 
sides  and  the  top 
and  bottom  are  now 
foreshortened  and 
their  horizontal  lines 
vanish  respectively 
in  VPl  and  VP2 
(Fig.  177). 

Selection    of    Sub- 
ject,  and   Use   of    the 
Picture  Plane. — It  will 
be  seen  that  the  room 
in  Fig.  174  is  the  same  as  in  the  previous  chapter.     The  differ- 
ence is  in  the  selection  of  subject-space,  and  in  the  consequent 

110 


Fig.  174 


ROOM    AT    ANGLES    TO    PICTURE    PLANE 


relation  of  the  subject-matter  within  that  space  to  the  picture 
plane.  Thus  in  the  last  study,  the  extreme  points  {x  and  y)  of  the 
back  wall  of  the 
room  are  equally 
distant  from  the 
picture  plane  as 
well  as  from  the 
eye  (A  in  Fig.  175). 
Therefore  we  could 
not,  without  ab- 
surdity, vanish  the 
lines  on  this  wall 
in  either  direction 
(A    in    Fig.     176), 


^      Picture 

Plane 

J 

\                z 

/ 

\               ° 

/ 

\              \- 

o 

/ 

\            ^ 

z 

\          ^ 

■ui                , 

/ 

\         ^ 

u             / 
•^          / 

\       < 

^        / 

\      1^ 

o      / 

\     t- 

A 

Fig.  175 


still    less   in   both 

(B    in     Fig.    176). 

The  only  way  to 

give   a  truthful 

impression  of   the 

back  wall  is  to  draw  it  in  its  true  shape  (C  in  Fig.  176),  as  was 

done  in  the  last  chapter. 

But  in  the  present  ex- 
ercise (Fig.  171)  part  of 
the  room  is  left  out.  The 
central  direction  of  seeing 
is  therefore  moved  to  the 
right,  and  with  it  the  pic- 
ture plane  is  turned  (B, 
Fig.  175).  Consequently 
the  wall,  xy,  which  was 
before  parallel  to  the  pic- 
ture plane,  now  recedes 
from  it.  Hence  the  height 
of  the  room  at  the  corner, 
being  farther  into  the  pic- 


FiG.  176 


111 


FREEHAND    PERSPECTIVE 

ture  and  from  the  picture  plane,  appears  less  than  at  the  right 
and  left;  and  all  horizontal  lines  on  both  walls  appear  to 
converge. 

In  this   drawing  the   rectangle   incloses   the    parts   of   most 
interest  and  cuts  off  the  awkward  outer  lines  (Fig.  177).    An 


Fig.  177 

alternate  selection  is  indicated  by  dotted  lines.  This  brings  us 
to  the  consideration  of  a  new  point,  namely:  — 

What  to  Include  in  a  Picture.  —  This  cannot  be  all  that  it  is 
possible  to  see  from  any  point,  for  the  head  can  be  turned  to 
see  all  parts  of  the  horizon  circle.  Such  a  view  when  painted 
forms  a  panorama,  which  is  a  continuous  cylindrical  picture 
surrounding  the  spectator. 

It  is  evident  that  the  legitimate  picture  must  include,  or  cut 
out  from  what  it  is  possible  to  see,  only  such  a  space  as  can  be 
perceived  by  the  eye  in  a  single  effort  of  seeing.  It  should 
leave  out  whatever  cannot  be  seen  without  turning  the  head, 
or  even  noticeably  moving  the  eyes.     It  is  generally  understood 

112 


ROOM  AT  ANGLES  TO  PICTURE  PLANE 


that  sixty  degrees  of  the  horizon  circle  is  the  most  that 
be  taken,  and  that  usually  thirty  degrees  is  better.  The 
that  the  greatest  dimension  of  the 
selected  space  (whether  height  or 
width)  shall  not  exceed  the  artist's 
distance  from  that  dimension. 

It  will  be  seen  by  the  diagram 
(Fig.  178)  that  this  is  equivalent 
to  not  exceeding  sixty  degrees  in 
the  picture.  It  is  also  apparent 
that  it  does  not  prevent  the  in- 
clusion of  objects  nearer  than  that 
greatest  width  if  they  come  into 
the  picture  space,  as  the  table  and 
part  of  the  rug  in  this  illustration. 

In  conclusion  it  should  be 
noted  : 

First,  —  Tlie  picture  plane  is  dif- 
ferent for  each  neiv  picture  selection. 

Second.  —  The  picture  angle 
should  not  he  over  sixty  degrees^  and 
is  letter  less. 


3.  the  pa[zt  selected  fob  a  picture . 
Fig.  178 


should 
rule  is 


118 


Chapter  XXXV 
FURTHER  STUDIES  OF  INTERIORS 

THE  student  should  copy  carefully  Fig.  179,  always  deter- 
mining the  level  of  the  eye,  and  locating  the  vanishing 
points  either  actually  or   mentally.     Note  that  the  em- 
phasis  of    contrast  and    interest  is  concentrated    on    the  old- 


i  ~  '  r<,5\ 


^ 


114 


Fig.  179 


FURTHER    STUDIES    OF    INTERIORS 


fashioned  desk  while  the  details  on  the  wall  beyond  it  are  very- 
quiet.  The  settle  comes  forward  more  (that  is,  has  more  con- 
trast and  emphasis)  than  the  wall,  but 
less  than  the  desk.  The  floor  is  pur- 
posely quiet  in  detail  to  aid  in  con- 
centration of  interest. 


Fig.  180 


Fig.  181 

(Figs.  180,  181, 
artistic  value  of 
principal    sets   of 


and 

the 


The  chair  in  the 
extreme  right  of 
the  fore-ground 
is  subordinated 
also. 

Original  Work.  —  Following  the 
above  copy  several  interiors  should  be 
drawn  from  the  place.  The  finder 
should  be  used,  and  thumb-nail  sketches 
182)  should  be  made  to  test  the 
selection.     Avoid    equal    angles    in  the 

should    be   drawn 


vanishing  lines.     If   both 

at  forty-five  degrees,  or  nearly  that,  th^ 

composition   would  have   a  stiff  effect. 

The   contrast   between   surfaces    turned 

away    much    and    others    turned    away 

little  is  generally  pleasing.  Should  it 
be  impossible  to  avoid 
equal  angles  of  van- 
ishing, relieve  the  stiff- 
ness by  using  details 
(as  the  sofa  and  win- 
dow in  Fig.  183)  that 

are  widely  different  in  effect.  The  rug, 
being  parallel  with  the  sofa,  assists  further  in 
overcoming  the  monotony.  It  is  undesir- 
able to  show  the  floor  and  ceiling  as  occupy- 
ing equal   space   in    the   picture;     and    it    is 

usually  better  not  to   show  both  floor   and   ceiling.     (Compare 

Fig.  182  with  the  same  subject  in  Fig.  181.)    A  subject  may 

115 


Fig.  183 


FREEHAND    PERSPECTIVE 


Fig.  184 


often    be    improved    by    cutting    it    with    a    different    margin 
(Fig.  185). 

Care  should  be  taken  to  place  the  principal  oblique  lines 
of  the  study  in  such  a  relation  to  the 
margin  or  inclosing  rectangle  that  they 
do  not  conspicuously  point  to  the  picture 
corners.  The  rectangle  which  bounds  a 
composition  is  an  orderly  conventional 
shape,  and 
hence  unno- 
ticeable,  leav- 
ing the  atten- 
tion  to  be 
concentrated 
on  the  picture. 
Lines  to  its  corners  direct  the  atten- 
tion there,  and  defeat  this  end. 

Vignetting.  —  It  is  not  necessary  p^^  jg^ 

that  an  inclosing  margin  should  always  be 

used.  The  drawing 
may  be  vignetted,  or 
blended  off  into  the 
white  paper  (Fig. 
186).  This  is  less 
easy  to  do  as  the 
tendency  of  the  ir- 
regular outer  edge 
is  to  contrast  sharply 
with  the  white  paper, 
detracting  from  the 
effect  of  the  more 
important  central 
parts.  The  edge  details  therefore  must  be  carefully  "  quieted," 
or  rendered  inconspicuous. 

Interesting  effects  are  often  produced  with  a  partial  margin 

116 


Fig.  186 


FURTHER    STUDIES    OF    INTERIORS 

line  (Fig.   187).     This  is   advisable  if  the   center  of   interest  is 
too  near  the  edge  of  the  paper  to  blend  off  well. 

So  many  principles  of  perspective  and  of  artistic  rendering 
are  included  in  the  drawing  of  interiors  that  they  form  a  most 
important  division  of  the  subject.  In  figure  compositions  they 
are  constantly  used,  especially  by  the  illustrator. 


Fig.  187 


117 


T 


Chapter  XXXVI 
A  CHAIR 

HE  chair  in  Fig.  188,  like  most  chairs  is  different  from 
the  stool  in  our  first  interior  study  (Ch.  XXXIII)  in 
that  the  sides  are  not  parallel  to  each  other  (see  plan, 

— .„ _^.- :_ Fig.  189).    Also 

t  the  seat  is  a 
trifle  lower 
in  the  back, 
so  that  lines 
B,  B  (Fig.  190) 
slope  down- 
ward toward 
the  back.  But 
the  chair  is 
symmetrical  on 
its  center  line 
(front  view, 
Fig.  189),  there- 
fore the  hori- 
zontal lines  in 
its  front  and 
back  are  par- 
allel, having  a 
common  van- 
ishing point  on 
the  eye  level. 

Drawing    the 
Chair.  —  After 

J    having  planned 

its  place  on  the 

118 


Fig.  188 


A    CHAIR 


L     I      \ 

PLAN  OF  5CAT 


paper  and  located  the  eye  level,  the  next  step  is  to  take  the  direc- 
tion of  the  front  and  side  of  the  seat  (lines  A  and  B,  Fig.  190). 
Leaving  line  A  for  the  present  we  carry  line  B  out,  obtaining 
VPl.  The  feet,  being  on  the  horizontal  floor,  fall  in  lines  vanish- 
ing to  the  eye  level  —  on  the  right  to  VPl,  and  on  the  left  in  lines 
E  and  F.  But  as  the  chair  sides  are  not  parallel,  line  E,  being 
turned  away  from  the  picture  plane  more  than  F,  appears  steeper, 
and  its  vanishing  point,  VP2,  is  nearer  ; 

than  VPS,  to  which  line  F  vanishes.  | 

The  near  side  of  the  seat,  line  A,  being 
in  the  same  vertical  plane  with  line  E 
(directly  over  it)  must  vanish  where  it 
will  cut  the  vanishing  trace  of  this  ver- 
tical plane,  which  is  a^  vertical  line 
through  the  vanishing  point  of  line  E. 
(Compare  the  roof,  p.  72.)  Carrying  out 
line  A,  therefore,  to  this  vertical  trace 
gives  its  vanishing  point,  OVPl,  to 
which  we  vanish  line  Gr,  which  is  parallel  with  A. 

In  the  same  way,  line  F,  vanishing  to  a  point  on  the  eye  level 
a  little  further  away  than  VP2,  gives  the  location  of  a  vertical 
trace  for  0VP2,  in  which  the  other  (invisible)  side  of  the  seat 
must  vanish.  The  crossing  of  these  invisible  seat  edges  (H  and  I) 
and  of  the  lines  C  and  F  aid  in  shaping  the  further  leg. 


raoNT  vitw 


Fig.  189 


Fig.   190 

As  the  seat  is  evenly  slanted,  0VP2  is  on  a  level  with  OVPl. 

All  edges  parallel  to  the  front  and  back  converge  to  VP2. 
The  curving  front  of  the  seat  and  the  curves  of  the  back  are 
drawn  so  their  ends  rest  on  a  line  to  VP2.     A  rocking  chair  is 

119 


FREEHAND    PERSPECTIVE 

simply  a  chair  (A,  Fig.  191)  placed  on  rockers  (B,  Fig.  191).  The 
weight  of  its  back  generally  tips  it  backward  when  at  rest,  hence 

the  usual  appearance  of  a  rocking  chair  is 
that  of  an  ordinary  chair  plus  its  rockers. 
In  B,  Fig.  191,  lines  A  and  B,  which  above 
are  actually  horizontal,  here  actually  in- 
cline downward  at  the  back;  and  must 
vanish  to  a  point  below  the  eye  level  (like 
A,  G  and  H  in  Fig.  190).  The  seat  and 
arms  are  more  foreshortened,  and  the  back 
less  erect.  But  the  horizontal  lines  C,  D, 
E,  F  and  their  parallels,  remain  actually 
horizontal,  and  vanish  to  the  same  point 
as  in  A.  The  directions  of  the  slanting 
back,  the  legs  and  the  arm  supports  should 
be  carefully  taken,  remembering  that  their 
ends  and  ornamental  details  (as  G,  H  and 
D)  fall  in  lines  to  VP2.  In  sketching 
straight  guide  lines  for  the  rockers,  recall 
that  they  are  farther  apart  in  front,  to  agree  with  the  sides  of  the 
seat.  The  lines  where  the  cylindrical  rungs  enter  the  legs  are 
actually  modified  circles,  hence  they  will  be  seen  as  shapes  modi- 
fied from  the  elliptical,  and  should  be  carefully  considered  and 
drawn. 

Following  this,  the  student  should  choose  and  draw  at  least 
one  other  piece  of  furniture.  Selection  should  be  made  of 
something  interesting  in  itself ;  that  is,  well  designed  and  con- 
structed, and  agreeable  in  association.  In  chairs  the  old-fash- 
ioned rush  or  splint  bottom  ones,  the  wooden  rockers  of  our 
grandmothers,  the  beautiful  examples  by  Chippendale,  Sheraton 
and  others  of  the  colonial  period,  and  good  examples  of  modern 
mission  shapes  may  be  mentioned  as  among  those  satisfactory 
for  study.  If  one  is  fortunate  enough  to  get  a  really  fine  old 
cradle,  it  is  a  most  instructive  subject,  as  is  an  antique  desk  or 
a  tall  clock. 


Fig.  191 


120 


T 


/- 


> 


k 


1 


r 


/ 


Chapter  XXXVII 

THE  HEXAGONAL  PLINTH  IN  TWO 
POSITIONS 

HE  student  should  draw  this  example  from   the  objects, 
following  the   directions   here   given;    and   should   then 
sketch  them  from  memory,  as  by  the  general  directions 
.  in  Chapter  XI. 
The  Model.  ^ 

This    can    be 

made  from  card- 
board according 

to   the  diagram 

(Fig.  193).     Tor     ,       '       \.__.^.,_.,„__^^' 

the    first    posi- 
tion   it    should 

be  placed  about 

three  feet  from     '  '^^^ 

the    eye    and 

nine  inches  be- 
low it,  and  with 

two    vertical     \  ^' 

faces  parallel 

with  the  picture 

plane. 

The    Greomet-  ,,  ,  ^    , 

ric  Hexagon. —      ]  "     .       |  \/ 

This  should  be 

constructed 

first.      Divide 

the    line    AB 

(Fig.    194)    in     \ --^^.^___^_._ ^ I 

halves,  drawing  f,^  192 

121 


FREEHAND    PERSPECTIVE 


■•^  h  -^ 


Fig.  193 


a  perpendicular  at  the  point  of  division.     Measure  the  distance 
AB,  taken   on   the   pencil,  from  B  against   the   perpendicular. 

Where  it  falls  (at  O)  will  be  one  corner 
of  an  equilateral  triangle  (AOB)  which 
will  form  one  sixth  of  the  desired  hexa- 
gon. Sketch  vertical  lines  of  indefinite 
length  from  A  and  B,  and  cut  them  by 
continuing  the  sides  of  the  equilateral 
triangle  (AO  and 
BO)  to  D  and  C, 
then  draw  DC.  The 
constructive  rec- 
tangle ABCD,  which 
we  have  now  com- 
pleted, will  be  al- 
ways essential  in 
drawing  the  perspective  of  the  hexagon. 
Its  diagonals  (A  and  B)  will  form  two  di- 
agonals of  the  hexagon,  and  its  center  (0) 
will  be  the  center  of  the  hexagon.  The 
other  diagonal,  EF,  is  drawn  through  O  parallel  to  the  rectangle 
ends,  half  on  each  side  of  the  center.  Its  length  should  be  tested 
by  that  of  the  diagonals  already  found.  The  other  four  sides 
complete  the  geometric  hexagon  ABCDEF. 

Drawing  the  Hexagonal  Top.  —  The  rectangle  ABCD  (Fig.  195) 
is  drawn  first.  Although  it  is  actually  nearly  twice  as  long  as 
its  width  it  will  be  found  to  appear  less  than  half  as  long.  The 
diagonals  of  the  rectangle  will  in  perspective,  as  in  the  geo- 
metric view,  form  two  diagonals  of  the  hexagon.  The  other 
diagonal,  EF,  is  set  off  on  a  line  of  indefinite  length  through 
their  crossing.  Looking  at  the  diagram,  it  is  observed  that  the 
sides  of  the  rectangle  (AC  and  BD)  and  its  middle  (0)  divide 
this  diagonal  (EF)  into  four  equal  parts  at  x,  0  and  y.  In  the 
perspective  drawing  (Fig.  195),  the  two  middle  fourths  (xO  and 
Oy)  are  seen  to  be  already  measured.     Since  these  fourths  are 

122 


Fig.  194 


HEXAGONAL    PLINTH    IN    TWO    POSITIONS 

all  equally  distant  from  the  picture  plane,  they  appear  equal,  and 
are  so  set  off  from  x  and  y.  The  hexagonal  top  is  completed  by 
drawing  its  last  four  sides. 

The  Thickness  of  the  Plinth.  —  The  front  face  of  this  thick- 
ness, being  parallel  to  the  picture  plane,  is  drawn  in  its  true 
shape.  The  lower  edges  of  the  other  two  faces  are  parallel  to 
the  receding  horizontal  edges  (AE  and  BE)  above  them,  and 
will  therefore  vanish  with  these  edges  respectively  to  VP2^and 
VPS.  Vertical  lines  downward  from  E  and  F  will  complete  the 
plinth. 


VPl 


VPI 


fVf  LEl/EL 


Fig.  195 


A  Test  for  Vanishing  Lines.  —  The  other  two  diagonals  and 
the  receding  back  edges  of  the  hexagon  are  in  reality  each 
parallel  to  one  or  the  other  of  the  two  sets  of  vanishing  lines 
just  drawn.  Therefore  when  carried  out  to  the  eye  level, 
they  should  meet  respectively  in  VPI  and  VP2  if  the  drawing 
is  correct. 

The  Hexagonal  Plinth  Slightly  Turned.  —  For  this  drawing  turn 
the  model  so  its  front  face  will  make  an  angle  of  thirty  degrees 
with  the  picture  plane  (B  in  Fig.  196).  Draw  the  constructive 
rectangle,  ABCD ;  taking  with  the  pencil  the  direction  of  the  front 
edge  and  of  the  imaginary  left  side  (AC  in  Fig.  196)^  of  the  rec- 
tangle.    Take  especial  pains  to  have  these  lines  correct  in  direction 

1  Corresponding  to  the  receding  edges  (2,  2)  first  drawn  in  sketching  the  cube. 

123 


FREEHAND    PERSPECTIVE 


and  length,  as  an  error  here  causes  a  particularly  unpleasant 
representation.  In  vanishing  the  back  edge  with  the  front  one  be 
careful  to  keep  it  also  tending  upward. 


CYE    LEVEL 


iPlOTVRtl  PlONC 
R-  PI-AM  or  HEXAGON  8£EN  IN  A. 


Fig.  196 


A.  Wrono.   Lines  I,  2,  3 

AND  4  DO  NOT  CONVERGE 
TO  THE    EYE  LEVEL. 


The  vanishing  point  (VP2)  of  these  front  and  back  edges 
is  so  far  away  that   their   convergence  will   naturally  be  only 

jestimated.  Hence  the  necessity  of  locating 
definitely  in  mind  what  cannot  he  seen  — 
that  is,  the  vanishing  point,  not  only  of 
these  two  lines,  but  of  their  parallels, 
the  diagonal  EF  and  the  lower  edge  of 
the  front  face. 

In  this  position  the  last  diagonal,  EF, 
is  in  perspective,  and  therefore  its  four 
actually  equal  divisions  will  appear  de- 
creasing in  size,  or  perspectively  equal. 
Consequently,  if  the  work  so  far  done  is 
correct  it  will  be  found  that  the  space 
xO  is  (almost  imperceptibly)  greater  than 
Oy^  because  a  little  nearer.  Hence  E:c  should  be  set  off  a  little 
larger  than  xO,  and  y¥  a  little  smaller  than  Oy.  Similar  cases, 
as  the  cylinder  top  in  Chapter  IV,  and  the  concentric  circles  in 
Chapter  XX  are  readily  recalled. 

Testing  the  Drawing.  —  The  test  used  for  the  other  drawing 
of  the  plinth  is  equally  effective  here.     It  is  more  needed  here, 

124 


B.  Wrong-,   lines  J,  2,3 

AND    4  ARE    TOO    &TEE.P. 

hexagon  appears  tiltep. 
Fig.  197 


HEXAGONAL    PLINTH    IN    TWO    POSITIONS 

since  the  vanishing  point  for  one  set  of  lines  has  not  been 
actually  found.  But  if  these  lines  have  been  carefully  thought 
out  as  to  direction,  the  errors  will  be  found  encouragingly 
slight. 

The  test  of  placing  the  eye  at  the  vanishing  point  (Ch.  XVH), 
to  sight  back  along  the  lines  which  should  converge  to  them,  is 
especially  applicable  here. 


125 


T 


<mM»"g"'""'«">'"' 


ni  rm 


INTERIOR  WITH  A  TILED  FLOOR 

HE  plan  of  this  floor  is  shown  in  Fig.  199.  As  drawn  in 
the  example,  the  foreshortening  of  the  tiles  is  proportioned 
to  that  of  parallel  surfaces,  such  as  the  receding  "  treads  " 

or  tops  of  the  steps. 
From  the  vanish- 
ing of  their  edges, 
we  may  judge 
these  treads  to  be 
foreshortened  about 
one  half.  Conse- 
quently lines  paral- 
lel to  the  staircase 
edges  (vanishing  in 
VPl)  must  be  fore- 
shortened as  much. 
The  tile  adjacent 
to  the  lowest  step 
is  the  best  to  begin 
with  (since  it  is  the 
same  distance  into 
the  picture).  As  with 
the  previous  hex- 
agons the  rectangle 
ABCD  is  drawn 
first.  And  since 
much  depends  on 
the  correctness  of 
this  first  rectangle, 
it  is  worth  while  to 


■iW   ■jatnuttf  iirii<».j 


J 


Fig.  198 


126 


INTERIOR    WITH    A    TILED    FLOOR 


take  especial  pains  with  it.  Observe  that  as  the  tiles  are  here 
placed,  it  is  the  width  (AD)  of  this  rectangle  which  is 
parallel  with  the  receding  staircase  edges. 
Thus  these  tiles  are  three  and  a  half  inches 
on  their  edges,  making  the  rectangle  width 
actually  half  the  height  of  a  seven-inch  step.  In 
the  drawing  this  width,  being  foreshortened  as 

much  as  the 


steps,  ap- 
pears only 
one  fourth 
as  wide  as 
the  height  of  the  step.  The 
actual  length  of  the  rectangle 
is  seen  by  the  diagram  to  be 
one  eighth  less  than  twice  its 


Fig.  199 


Fig.  200 


width;   or  (what   is  the  same  thing)  one    eighth  less   than   the 

height   of  the   step.      Being  slightly  turned   e^pView  of^tep 

away,  it  will  appear  a  very  little  shorter  in 

comparison  than  that.      In  this  case  it  was 

made  one  sixth  less  than  the  height  of  the 

step. 

It  will  be  readily  seen  how  if  the  rectangle 
proportions  are  right,  the  lines  of  this  first 
hexagon,  when  carried  out  forward  and  back,  will  give  points  for 
the  other  tiles,  making  them  fall  harmoniously  into  their  proper 
perspective. 


Fig.  201 


127 


Chapter  XXXIX 
THE  HEXAGONAL  PRISM  AND  FRAME 


T 


HIS  exercise  may  be  drawn  from  the  objects,  if  they  are 
at  hand.  If  they  cannot  readily  be  had,  the  drawing 
juay  be  made  from  these  directions,  using  the  cardboard 

plinth  made  for 
Chapter  XXXVII. 
The  objects  should 
then  be  drawn 
from  memory. 

The  Prism.  —  Be- 
ing the  simpler  to 
draw,  this  object 
should  be  taken 
first,  though  it 
should  be  placed 
at  the  bottom  of 
the  sheet,  on  ac- 
count of  its  greater 
horizontal  dimen- 
sions. 

This  model  is 
eight  inches  long, 
and  the  diameter 
of  its  hexagonal 
bases  is  four  inches. 
It  is  placed  so  that 
the  bases  make 
angles  of  sixty  de- 
grees with  the  pic- 
ture plane.     The 


Fig.  202 


128 


THE    HEXAGONAL    PRISM    AND    FRAME 


Fig.  203 


dotted  lines  in  Fig.  203  show  how  the  cardboard  plinth  may  be 
placed  in  the  same  position  as  a  help  in  study. 

The  Nearest  Vertical  Hexagonal  Base.  —  Sketch  the  construc- 
tive rectangle  previously  used,  noting  that  its  width  is  fore- 
shortened as  much  as  the  most  foreshort- 
ened side  of  the  cube  in  Chapter  XVII. 
Set  off  the  third  diagonal  of  the  hexagon 
(EF)  perspectively  on  a  line  vanishing  to 
VPl  through  the  rectangle  center  (as  on 
page  124). 

The  Long  Edges  of  the  Prism.  —  Take 
the  direction  of  the  nearest  upper  edge,  giving  VP2.  Vanish 
the  other  long  edges  with  it,  and  set  off  on  the  one  first  drawn 
its  apparent  length  (AGr).  This  can  be  easily  estimated  by  re- 
calling the  cube.  That  is,  its  actual  length  as  given  is  twice  that 
of  the  diameter  of  its  base  (the  near  vertical  line  AB).  AGr  will 
therefore  be  as  long  as  two  cubes  placed  side  to  side.^ 

The  Further  Base.  —  For  the  horizontal  top  edge  of  the  further 
base  draw  a  line  parallel  to  the  same  line  in  the  near  one  (that 
is,  vanishing  in  VPl),  which  gives  GH.  For  the  corner  corre- 
sponding to  B  in  the  near  base  drop  a  vertical 
from  Gr,  giving  J.  The  upper  oblique  edge 
(GI)  is  parallel  to  AF  in  the  near  base,  and 
their  vanishing  point  is  OVPl,  vertically 
above  VPl.  Another  oblique  line,  from  the 
nearest  point,  I,  to  the  lowest,  J,  completes 
the  prism. 

The  Hexagonal  Frame.  —  This  model  is  three 
inches  on  a  side,  and  is  one  inch  square  in  sec- 
tion (Fig.  204).  It  stands  on  one  rectangular 
face  with  its  hexagonal  faces  at  an  angle  of 
thirty  degrees  with  the  picture  plane  (Fig.  205) .  For  the  outer 
hexagon   and    the   outer  thickness    proceed    as   in   the    prism. 


Face  view  or  fpame 

Section  of  fi?ame 
Fig.  204 


XVI. 


This  method  is  given  ia  a  slightly  different  form  under  Solutions  of  Problems,  Chapter 
9  129 


FREEHAND    PERSPECTIVE 

For  the  inner  hexagon  we  may  first  study  the  actual  shape  in 
Fig.  204,  where  it  is  seen  that  the  vertical  frame  thickness  can 
be  conveniently  carried  across  to  measure  it  at  C^  on  the 
nearest  vertical,  BC.  We  know  this  thickness  to  be  one  inch, 
which  is  more  than  one   sixth  and   less   than   one  fifth  of  the 

vertical  CB.  If  one  fifth  of  CB  (CO  in 
Fig.  205)  be  found,  and  three  quarters  of 
this  {Cy)  be  taken,  it  will  serve  the  purpose. 
Mark  the  same  distance  from  B  up  (point 
2).  From  these  points  {y  and  b)  vanish  the 
horizontal  lines  of  this  inner  hexagon  with 
their  parallels  to  VPl.  The  corners  of  the 
inner  hexagon  are  on  the  diagonals  of  the 
outer  one,  so  the  crossings  of  the  diagonals 
by  these  two  vanishing  lines  give  four  corners  of  the  inner 
hexagon  (1,  2,  3,  and  4).  From  2  an  oblique  line  vanishes  down- 
ward with  its  parallels  to  0VP2,  marking  point  5  on  the  hori- 
zontal diagonal  EF.  Another  from  3,  also  vanishing  in  0VP2, 
is  drawn  from  3  upward  to  cut  EF  in  point  6.  Lines  from  6  to 
1  and  from  2  to  5,  complete  the  inner  hexagon ;  and  should,  if 
the  drawing  is  correct,  vanish  with  those  parallel  to  them  in 
OVPl. 

The  inner  edges  of  the  thickness  which  are  visible  vanish 
from  4  and  5  to  VP2,  being  parallel  to  the  outer  thickness  edges. 
For  the  visible  part  of  the  further  inner  hexagon,  a  line  of  the 
near  inner  hexagon,  as  4-3,  may  be  carried  "  around  the  corner  " 
and  back,  as  in  the  square  frame  (Ch.  XXIV).  From  the  point 
where  this  vanishing  line  (7-8)  cuts  the  line  from  4  to  VP2,  an 
edge  (8-9)  vanishes  with  its  parallels  (CF  and  others)  to  OVPl. 
This  point  is  even  further  away  than  0VP2,  so  that  the  con- 
vergence of  its  vanishing  lines  must  be  slighter.  Where  line  8-9 
crosses  the  one  from  5  to  VP2  it  meets  the  last  visible  edge  of 
this  back  inner  hexagon,  a  line  vanishing  in  0VP2. 


130 


Chapter  XL 


THE  TRIANGULAR  PRISM  AND  FRAME  — 
PROBLEM  FOR  ORIGINAL  STUDY 


THE  MODELS.  —  The  prism  is   eight   inches   long,  and   its 
triangular  ends  are  four  inches  on  a  side.     The  frame  is 
six   inches   on   a   side,  and   one   inch  square  in   section. 
See  diagrams.  Fig.  206. 


A    TEIANQULAR    FACE 
OF    FPAME 


B.  5IC)E  VIEW 
OF  FPAME 


Positions.  —  The  objects  are 
placed  four  feet  away,  and  one 
below  the  eye,  and  are  separated, 
as  were  the  models  in  the  last 
chapter.  The  prism  rests  on 
one  long  face,  with  its  long  edges 
making  angles  of  thirty  degrees 
with  the  picture  plane.  The 
frame  rests  on  a  rectangular  face 
with  its  triangular  faces  at  thirty 
degrees  with  the  picture  plane. 

Arrangement  on  the  Sheet.  —  The  drawings  are  to  be  placed  on 
one  sheet.  The  position  of  the  paper  (whether  with  its  long 
edges  horizontal  or  vertical)  and  the  placing  of  the  drawings 
on  the  sheet  must  be  such  as  to  produce  the  most  agreeable  and 
satisfactory  effect. 


C    END  OF  PPISM 


d    51 de  view  of  pei3m 
Fig.  206 


131 


Chapter  XLI 
THE  STUDY  OF  PARALLEL  PERSPECTIVE 


F 


Fig.  207 


ROM  Chapters  XXXIV  and  XXXV  it  is  seen  that  inte- 
riors follow  the  law  of  the  cube.  This,  however,  leads  to 
what  may  seem  an  inconsistency.     Why,  it  may  be  asked, 

does  the  table  in  Fig.  207  differ  from 
the  cube  in  Fig.  208^  In  the  cube 
B  was  made  shorter  than  A  because 
farther  into  the  picture.  But  in 
the  table  B  was  not  drawn  shorter 
than  A. 

The  answer  to  this  is  that  the 
table  was  not  studied 
alone,  as  was  the  cube. 
It  was  part  of  a  pic- 
ture in  which  the  dominant  part  (the  back  of  the 
room)  was  parallel  to  the  picture  plane.  Having 
drawn  the  side  of  the  table  parallel  to  the  sides 

of  the  room,  it  is  absurd  (Fig.  209) 
to  draw  its  end  otherwise  than 
parallel  with  the  back  of  the  room. 
Fig.  207  satisfies  the  eye  and  gives 
a  true  impression  of  the  room  and 
its  contents.  Could  the  room  be 
erased,  leaving  the  table  alone,  it 
would  present  the  error  shown  in 
Fig.  210.  Here  it  forms  the  whole 
picture  and  its  picture  plane  makes 
an  angle  with  its  ends  (see  plan  in  Fig.  210),  hence  it  must  be 
drawn  as  below. 

132 


Fig.  208 


THE  TABLE    I5WB0N0 

Fig.  209 


STUDY    OF    PARALLEL    PERSPECTIVE 


COCTSECTION 


Fig. 210 


It  is  undeniable  that  in  Fig.  207  B  is  farther  from  the  eye 
than  A.  But  since  drawing  that  corner  smaller  produces  the 
false  impression  seen  in  Fig. 
209,  we  are  guided  by  the  dis- 
tance of  these  points  not  from 
the  eye,  but  from  the  picture 
plane.  The  picture  plane  simply 
forms  the  best  means  of  attain- 
ing our  fundamental  object  —  a 
truthful  representation.  Hence 
the  necessity  of  determining 
the  limits  of  the  picture  (Fig. 
211)  and  of  clearly  fixing  in  mind  reSviSn  orp™E  plane 

TO  TABLE.  WHEN   TABLE 

the   central  direction  of   seeing         '^  ^'-°''^ 
and  the  picture  plane. 

Under  some  conditions,  sur- 
faces may  even  be  drawn  in 
their  true  shape  when  not  quite 

parallel  to  the  picture  plane.  In  Fig.  211 
none  of  the  vertical  surfaces,  as  A,  B,  and  C, 
are  exactly  parallel  with  the  picture  plane. 
In  Fig.  212  this  is  shown  by  the  conver- 
gence of  the  lines  at  right  angles  to  these 
surfaces ;  their  vanishing  point  being  a  little 
out  of  the  center  of  the  picture.  Yet  if 
all  these  vertical  surfaces  are  drawn  in  per- 
spective (A  in  Fig.  212)  the  result  is  mis- 
leading or  impossible,  and  the  eye  protests. 
But  the  drawing  is  perfectly  satisfactory  in 
B,  Fig.  212. 

A  convincing  illustration  of  dominant 
surfaces  parallel  to  the  picture  plane  is  the 
familiar  form  of  a  bureau.  With  an  un- 
broken top  (A  in  Fig.  213)  it  is  easily  drawn  like  the  book 
and  cube.     If  now  the  middle  of  the  upper  drawer  is  cut  out,  the 

133 


'i^''^^/'^//////'/. 


PLAN  OF 
INTERIOR 
jHOWN    IN 

GuRE  ais.. 


Fig.  211 


FREEHAND    PERSPECTIVE 


A    UntpuE    drawing 

OF  VIEW    FROM    X,  IN  P(./\N 


B    CORRECT     DRAWING 
or  TME    SAME   VIEW. 


Fig.  212 


remaining  small  ones  are  seen  to  occupy  positions  similar  to  that 
of  the  table  in  Fig.  1. 

The  Street.  —  The    street  is   another  example   of  these   con- 
ditions.    Viewed  from  the  middle  of  a  crosswalk   {x  in  plan, 

Fig.  214)  the  fronts 
of  the  houses  pre- 
sent to  the  beholder 
a  perspective  like 
that  of  the  interior 
in  Fig.  207.  They 
vanish  to  the  center 
of  the  picture,  and 
surfaces  at  right 
angles  to  them  are 
drawn  in  their  true 
shape.  This  is  done 
even  if  the  con- 
vergence is  not 
toward  the  exact  middle  of  the  picture  (Fig.  215),  provided  it  does 

not  fall  to  right  or  left  of  the  house  fronts.  

In  this  view,  although  the  beholder  has 
passed  on  to  y,  the  conditions  are  still 
like  those  of  the  interior  in  Fig.  211. 

But  if  instead  of  using  both  sides 
of  the  street  for  our  picture,  we  choose 
one  of  the  corners,  the  picture  plane  for 
this  forms  a  different  angle  with  the 
principal  surfaces  (Fig.  214),  and  the  view 
must  be  drawn  as  shown  in  Fig.  216. 

It  may  therefore  be  concluded,  that 
171  any  picture  having  a  domhiant  imrt 
parallel  with  the  picture  plane  and  conse- 
quently clraivn  in  its  true  shape^  all  por- 
tions of  that  picture  which  are  parallel  tvith  the  picture  plane  must 
also  he  drawn  in  their  true  shape. 

134 


i^^^crr 


■<p 


.-4P- 


#     J ^^ 


Fig.  213 


STUDY    OF    PARALLEL    PERSPECTIVE 


Also,  even  such  dominant  parts  as  are  not  quite  parallel 
with  the  picture  plane  must 
be  drawn  in  their  true  shape 
in  certain  cases  ivliere  draiving 
them  in  perspective  produces 
false  or  misleading  results. 
Finally,  it  is  of  great  import- 
ance to  include  in  the  pic- 
ture only  what  can  easily  he 
^een. 

Parallel  perspective,  as  ■^^^^ 
work  under  such  conditions 
is  called,  involves  no  depart- 
ure in  principle  from  free- 
hand perspective  in  general. 
It  is  merely  an  adaptation 
of  perspective  methods  to 
certain  conditions  in  the 
subject.^ 

Space  has  been  given  here 
to  a  somewhat  extended  con- 
sideration   of    the    subject. 


Plan  OF  the 

^TEEET  6HOWN 
IN   PEESPECTIVE 


Fig.  214 


-Ti 


VIEW   or  JTEEET    FBOM      Y. 

Fig.  215 
1  The  terms  "  parallel  "  and  "  angular  "  perspective,  though  used  for  lack  of  better  ones, 


are  therefore  far  from  satisfactory. 


135 


FREEHAND    PERSPECTIVE 

because  the  confusion  concerning  it  that  frequently  exists  is 
deemed  unnecessary.  It  has  been  found  that  students  may  be 
easily  led  to  distinguish  when  such  conditions  are  present,  after 
which  there  is  no  difficulty  in  dealing  with  them. 


view  of  coenee  fbom  y 
Fig.  216 


136 


Chapter  XLII 

A  STREET  FROM  THE  PHOTOGRAPH 

THIS  exercise  (Fig.  217)  may  be  drawn  first  if  judged  best, 
noting   carefully  the  changes  made   in   rendering   from 
the  photograph  shown  in  Fig.  218.     The  student  should 
then  select  a  print 


street    and 


drawing 


of    a 
make    a 

from  it.  All 
sketches  should 
be  thoroughly  | 
thought  out,  hav- 
ing the  level  of 
the  eye  carefully 
placed,  and  all  the 
vanishing  points 
located,  either 
actually  or  men- 
tally.  It  will 
probably  be  neces- 
sary to  correct 
some  distortions 
of  the  camera  (see 
Ch.  XLIII). 

This  drawing 
should  be  followed 
by  studies  from  the 
street  and  from 
memory.  Notes 
and  sketches  made 


137 


Fig.  217 


A    STREET    FROM    THE    PHOTOGRAPH 

by  the  student  at  the  place  chosen  may  be  used  to  help  this 
memory  work. 


Fig.  218 
Entrance  to  the  Ponte  Vecchio,  Florence 


188 


Chapter  XLIII 


EXCEPTIONS  TO  THE  USE  OF  THE  FLAT 
PICTURE  PLANE 


IT  will  be  observed  that  in  photographs  the  circular  tops  of 
columns  near  the  edges  of  the  picture  often  appear  as  slant- 
ing ellipses  (Fig.  219).  And  all  who  have  an  acquaintance 
with  mechanical 
perspective  will 
recall  that  in  cer- 
tain problems  the 
ellipses  of  cylin- 
ders do  not  work 
out  at  right  an- 
gles to  the  axis 
(Fig.  220).  While 
the  eye  sees  ob- 
jects pictured  on 
the  inside  of  the 
spherical  eyeball, 
the  camera  forms 
its  pictures,  and 
mechanical  per- 
spective projects  its  problems  on  a  flat  surface.  Therefore  the 
camera  cannot  wholly  reproduce  objects  as  seen  by  the  eye, 
and  certain  results  obtained  by  mechanical  perspective  are 
untrue  representations.^ 

^  The  photographic  error  has  been  recognized,  and  a  camera  is  now  made  in  which  a 
clockwork  attachment  brings  each  part  of  the  plate  in  turn  directly  facing  the  part  of  the 
subject  it  is  to  receive,  and  gives  horizontal  ellipses  to  columns  wherever  placed  in  the 
picture. 

139 


Fig.  219 
Cloisters  of  the  Monastery  of  San  Martino,  Naples 


FREEHAND    PERSPECTIVE 


As  for  mechanical  perspective,  though  useful  in  many  cases, 
it  has  sometimes  obscured  the  real  aim  of  representative  drawing. 

It  has  even  been  taught  that  the 
flat  picture  plane  should  be  used 
for  all  representative  work  as  in 
mechanical  perspective,  logically 
to  the  end,  regardless  of  any  pro- 
test of  the  eye  as  to  its  results. 
To  this  error  it  is  sufficient  reply 
to   say  that  the  aim  of  freehand 


- 

^^=- 

^ 

— 

.    r 

1 I 

c1l 

f| 

^ 

1 

1 

f 

1 

t" 

Jit* 

^ 

Cylinders  as  found  by  mechanical  perspective, 
Side  cvlindebs  vntsue  as   REPREseivTATiONa. 


Fig.  220 


perspective  is  the  draiving  of  objects 
as  they  appear;  and  that  the  eye  never  sees  a  column  as  in  Fig. 
219,  nor  a  cylinder  as  the  outer 


ones  in  Fig.  220.  When,  there- 
fore, the  use  of  the  flat  picture 
plane  produces  an  untrue  draw- 
ing, it  is  evident  that  an  excep- 
tion must  be  made  in  that  case. 

The  Cylindrical  Picture  Plane. 
—  Looking  at  Fig.  220,  we  find 
that  the  middle  cylinder,  which 
does  appear  right  to  the  eye, 
extends  equally  each  side  of  the 
central  direction  of  seeing,  so 
that  the  picture  plane  is  parallel 
to  the  apparent  breadth  of  the 
cylinder.  By  drawing  the  other 
cylinders  as  if  each  had  such  a 
central  direction  of  seeing  and 
such  a  picture  plane  of  its 
own  (A,  Fig.  221)  a  result  is 
obtained  that  appears  true  to 
the  eye  (B,  Fig.  221). 

In   other  words,   cylindrical 


A.  Plan 
Showing 

Vi>E  OF5PEC- 
JAL    PICTURE 
PLANES- THE 
EQUIVALENT  OF 
THE  CylINDRi- 
CAU    PICTURE. 
PLANE  . 


EYE 


objects,  however  placed,  should  be 


B.  Appearancp  of  group  at  a, drawn  by 

\}i>e    OF  special -THATiS,    CYHNDRlCflU,  PLANED 

FiQ.  221 


140 


EXCEPTIONS    TO    FLAT    PICTURE    PLANE 


^  "  Side  viev/ 

'  Showing  the  u4E  or 

PICTURE    PLANES     INCLINED 
FROM  THE    VERTICAL-THftr    15, 

THE  Sphericau  picti;re  pw\n&. 
Fig.  222 


drawn  as  if  for  those  objects  alone,  the  picture  plane  was  bent  or  rolled 
into  a  cylindrical  picture  plane.  But  this  does  not  apply  to  the 
straight-line  portions  of  the 
picture  (as  the  block  in  Figs. 
220  and  221),  nor  to  the  plac- 
ing of  the  cylindrical  parts,  nor 
to  their  height.  These  must 
be  determined  in  the  ordinary 
way,  by  using  the  flat  picture 
plane.  We  onlj^  abandon  the 
flat  picture  plane  where  we 
cannot  otherwise  produce  a 
representation  which  the  eye 
will  accept  as  true. 

The  Spherical  Picture  Plane.  —  Another  exception  occurs  in  a 
vertical  direction.     Thus,  the  only  outline  that  will  truthfully 

represent  a  ball  to  the  eye  is  a 
circle.  To  obtain  that,  we  must 
regard  its  special  central  direction 
of  seeing  as  directed  to  its  middle, 
not  only  from  side  to  side  (as  in 
case  of  the  cylinder),  but  from 
top  to  bottom  also.  If  the  ball  is 
above  or  below  the  eye  therefore, 
its  special  picture  plane  is  slanted 
accordingly  (Fig.  222).  In  this 
case  the  picture  plane  (again /or 
such  ohjecfs  alone)  may  be  called 
a  Spherical  picture  plane. 

An  example  of  its  application 
is  the  case  of  a  model  posed 
higher  than  the  student  who  is 
drawing  (Fig.  223).  The  head  is 
foreshortened  vertically,  and  the  forehead  appears  smaller  in  pro- 
portion than  the  lower  and  nearer  features.    At  the  same  time  the 

141 


•''^•^iDI§J^ 


Fig.  223 


FREEHAND     PERSPECTIVE 

window  beyond  that  model  is  drawn  on  the  usual  flat  picture 
plane;  that  is,  with  its  vertical  lines  vertical,  as  always. 

These  distinctions  will  be  found  not  only  necessary,  but 
natural  and  easy  to  make ;  ^  especially  if  care  is  taken  to  include 
in  the  picture  space  only  what  the  eye  can  see  ivithout  noticeably 
moving  the  eyehaUs  (Ch.  XLI).  The  picture  plane  should  be 
regarded  as  limited  to  what  will  cover  this  selected  space,  and  we 
have  no  concern  with  what  lies  outside  of  that. 

When  working  from  a  photograph  therefore,  as  must  often  be 
done,  such  camera  distortions  as  the  columns  in  Fig.  219  should 
be  corrected  to  agree  with  what  is  pictured  by  the  eye.  And  in 
freehand  work  only  such  truths  of  mechanical  perspective  should 
be  used  as  produce  results  which  the  eye  confirms  as  true  repre- 
sentations. Where  the  eye  and  a  train  of  reasoning  are  in 
conflict,  the  reasoning  should  be  scanned  for  errors.  Unless  a 
drawing  looks  right,  it  may  safely  be  pronounced  not  right.  It 
may  look  right,  and  still  be  wrong ;  but  if  the  eye  refuses  to  be 
satisfied,  it  is  certainly  wrong. 

1  So  natural  and  easy,  in  fact,  that  space  for  this  explanation  is  hardly  needed,  except  to 
guard  against  false  reasoning  in  the  subject. 


142 


Chapter  XLIV 
SHADOWS 

WHILE  it  is  unnecessary  for  the  mastery  of  freehand 
sketching  to  study  this   subject   exhaustively,  there 
are  a  few  simple  facts  which  have  been  found  funda- 
mentally useful  in  practice,  and  which  may  be  easily  understood. 


Fig.  221 


To  that  end  the  student  should  follow  these  explanations 
carefully,  making  experiments  and  sketches  as  needed.  He 
should  then  compose  and  draw  a  group  similar  to  Fig.  224, 
also  should  make  other  studies  involving  the  use  of  the  truths 
here  developed. 

Light  may  be  regarded  as  composed  of   an  infinite  number 
of  rays.     From  a  lamp  they  extend   outward   in   all  directions, 

143 


FREEHAND    PERSPECTIVE 


forming  what  may  be  called  a  sphere  of  light.     The  shadow  of 

the  apple  on  the  right  of  the  lamp  in  Fig.  225  extends  toward  the 

right;  that  of  the  book 
on  the  left  in  an  almost 
opposite  direction.  The 
sun,  on  the  contrary,  is 
so  much  larger  than  the 
earth,  and  its  rays  Lave 
traveled  such  an  incon- 
ceivable distance,  that  to 
us  they  are  parallel,  as 
are  the  paths  of  fall- 
Fi«-  225  ing  raindrops.     Fig.  226 

illustrates  this  familiar  truth.     (This,  of  course,  is  also  true  of 

moonlight.)     There  are  therefore  two  classes,  of  shadows:   those 

cast   by  the   sun,  and  those 

produced  by  near  light,  as  a 

lamp.     Under  those  formed 

by  the  sun   may  be  studied 

first: 

Small  Objects  in  a  Room.  — 

If  a  shadow  box  be   placed 

near  and    a   little    back   of 

the   window,^   as    shown  in 

Fig.   227,   the   shadow   edge 

(A-6)   cast   by  the   vertical 

box  edge  AB  will  lie  on  the 

floor  of  the  box  in  a  line  actually  parallel  to  that  of  the  vertical 

hat-pin.     The  shadow  of  a  vertical  vase  (Fig.  228)  also  casts  a 

shadow  in  the  same  direction.     (That  is,  line  C,  the  shadow  of 

its  vertical  axis  will  be  parallel  with  the  shadows  of  the  vertical 

line  AB  on  the  box  floor,  and  on  the  horizontal  book  cover.)    The 

1  In  this  case  the  window  is  larger  than  the  box  ;  so  that  as  far  as  the  box  is  concerned  the 
rays  of  hght  are  parallel.  As  will  be  seen  later  in  this  chapter,  the  diffused  light  from  a  win- 
dow causes  radiating  shadows  in  the  room  itself. 

144 


Fig.  226 


SHADOWS 


shadow  (F^^)  of  the  vertical  book  corner  (FD)  will  be  parallel 
with  these  lines.  If  we  push  the  box  back  or  forward 
they  all  change  direc- 


EYE    LEVEL 


tion,  becoming  more 
nearly  parallel  with 
the  picture  plane 
as  the  box  moves 
forward,  and  vanish- 
ing more  steeply  if 
we  put  it  further 
back  of  the  window. 
But    they  always  Fig.  227 

remain  actually  parallel  to  each  other. 

In  the  same  way  we  see  in  Fig.  227  that  the  shadow  of  the 
horizontal  edge  BH  and  of  a  horizontal  hat-pin  (EF)  are  actually 
parallel  to  each  other,  on  both  the  back  and  the  floor  of  the 
box.  Also  in  Fig.  228  the  shadow  (b-e)  falling  on  the  hori- 
zontal book  cov- 
er from  the  hori- 
zontal box  edge 
(BE)  is  parallel 
to  the  shadow 
(d-g)  falling  on 
the  horizontal 
box  surface  from 
the  horizontal 
line  Da. 

Wemaythere- 
FiG.  228  fore  say  that  the 

shadows'  of  actually  parallel  lines  ivill  lie  actually  parallel  to  each 
other  on  the  same  surface  or  on  surfaces  parallel  with  each  other. 

But  in  perspective,    parallel   lines   vanish;    and  if  they  are 

horizontal  lines  they  vanish  to  the  level  of  the  eye.     We  should 

therefore  expect  these  parallel  shadows  to  also  vanish  thus,  and  we 

find  they  do  vanish,  in  the  same  manner  as  any  lines  or  objects. 

10  145 


FREEHAND    PERSPECTIVE 


Fig.  229 


We  next  observe  (in  Fig.  228)  that  the  vertical  vase  casts 
a  vertical  shadow  on  the  vertical  back  of  the  box.  Then  we 
recall  that  the  horizontal  edges  BE  and  DGr  cast  on  horizontal 

surfaces  horizontal 
shadow  edges  h-e  and 
d-g.  As  these  edges 
vanish,  their  shadows 
vanish  with  them  to 
the  same  point,  as  any 
parallel  lines  would. 

It  thus  appears  that 
when  the  receiving  sur- 
face is  pm'allel   to    the 
object  or  line  casting  the 
shadow,  the  shadow  will  also  he  parallel  to  the  object  or  line. 

This  brings  us  to  consider  how  to  find  the  extent  of  shadows. 
If  the  shadow  box  is  lowered  from  its  usual  place  on  the  table 
to  the  floor,  the  shadows  will  be  found  shorter  (Fig.  229).  The 
light,  falling  more  steeply,  cuts  off  the  shadows  nearer  the 
objects.  When  the  box  is  lifted  back  to  the  table  (Fig.  228)  the 
shadows  will  be  seen  to  lengthen. 
With  a  light  ruler  or  ''  straight- 
edge"  (Fig.  230),  take  the  actual 
direction  from  the  hat-pin  top  (D) 
to  its  shadow  (d)  on  the  floor  of 
the  box.  Keeping  the  ruler  in  the 
same  actual  direction,  move  it  to 
the  left  till  it  grazes  the  top  corner  (B)  of  the  box.  It  will  be 
found  also  to  mark  the  shadow  (h)  of  point  B. 

The  evident  truth  is  that  the  direction  of  the  light-ray  from  any 
point  in  the  ohject  7narhs  the  same  point  in  its  shadow.  Therefore 
to  find  the  shadow  of  any  point,  as  of  the  other  hat-pin  head 
(E,  Fig.  227)  we  have  only  to  draw  the  light-ray  from  E  to  where 
it  strikes  the  receiving  surface. 

But  since  in  this  case  the  window  is  nearer  than  the  box,  the 

146 


Fig.  230 


SHADOWS 


light-rays  are  receding  slightly,  hence  must  appear  to  converge 
a  little,  like  any  parallel  receding  lines.  Therefore  to  draw  them, 
we  first  find  their  vanishing  point.  Imagine  one  of  these  rays 
(as  D-^,  Fig,  227)  dropped  vertically  to  the  floor  of  the  box  (as 
the  gable  edge  in  Ch.  XXII  was  dropped).  It  would  then  lie 
in  the  horizontal  shadow  line  (C-^)  directly  under  it,  and  would 
vanish  in  VPS.  When  lifted  again  to  its  former  oblique  position 
(D-^)  its  vanishing  point  would  have  moved  down  in  a  vertical  line 
from  VPS,  and  become  OVPl.  All  other  light-rays  in  this  illustra- 
tion (as  Bb  and  E-e)  appear  to  converge  to  this  vanishing  point. 

Hotv  to  find  where  the  light  ray  mid  the  receiving  surface 
meet  is  the  iiext  consideration.  In  the 
case  of  D-^  we  had  a  vertical  line,  DC, 
and  its  shadow,  C-^,  cut  by  the  light-ray, 
T>-d.  From  E  we  can  imagine  a  vertical 
line,  similar  to  DC,  dropped  to  the  floor 
of  the  box.  (We  can  find  where  this 
vertical  line  will  touch  the  floor  by  a  vertical  from  F  to  the  box 
edge  MJ  at  point  K,  and  a  vanishing  line  from  VPl  through 
that  point,  K,  to  cut  the  vertical  from  E,  in  L.)  From  L  a  line 
actually  parallel  to  the  shadow  Q-d  (vanishing  in  VPS)  cuts  the 
light-ray  in  the  desired  point,  e.  This  is  essentially  the  way  most 
shadow  points  are  found:  — hy  a  vertical  line  from  the  point  on  the 
object  to  the  receiving  surface^^an^  from  that  a  shadow  line  on  the 
receiving  surface  to  cut  the  light-ray.  In  other  words  we  pass  an 
imaginary  vertical  plane  through  the  point  and  the  light-ray. 
Fig.  2S1  shows  a  simple  application  of  these  principles.  When 
the  cube  has  been  drawn,  the  direction  and  length  of  the  shadow 
edge  Ba  maybe  assumed  (or  taken  if  drawing  from  the  object). 
This  gives  the  direction  of  the  light-ray  A-a.  The  shadow  a-c 
vanishes  with  AC  till  cut  by  the  light-ray  from  C,  and  c-d 
vanishes  with  CD.  These  points  can  be  tested  by  vanishing  the 
light-rays  from  the  other  corners  to  OVPl,  thus  completing  the 
several  vertical  planes  as  above  mentioned. 

The  shadow  of  the  left  horizontal  box  edge  (BH)  falls  partly 

147 


FREEHAND    PERSPECTIVE 


ou  the  back  of  the  box  in  a  slantmg  Hue  which  may  be  thus  deter- 
mined. Take  the  book  out  (Fig.  227),  when  it  will  be  seen  that 
the  near  part  of  the  shadow,  beginning  at  h,  vanishes  on  the  floor  of 
the  box  to  yPl  till  it  reaches  the  box  edge  in  point  i.  The  shadow 
must  start  at  H,  hence  H-i  is  the  line  in  question.  Many  shadow 
lines  can  be  found  thus, — 1)1)  locating  any  two  points  in  the  line. 

Looking  again  at  the  shadow 
of  the  vase  on  the  back  of  the 
box,  we  observe  that  the  shadow 
of  its  horizontal  circular  top  f  all- 


^.■5HOWiNOr  A 
Veeticau  plane 
at  eioht  anoue5  to 

CENTER  or^hAPOW 


Perspective 

or  ABOVE. 


FACE  VIEW   OF 
VERTICAL  PART 
OF    5HAPOW. 


Fig.  232  Fig.  233 

ing  on  the  back  of  the  box  is  not  a  horizontal  curve.  To 
understand  this,  we  will  begin  with  the  shadow  of  any  vertical 
cylindrical  object  on  a  horizontal  surface,  as  in  Fig.  232.  It  will 
be  found  actually  symmetrical.  In  perspective  it  will  be  fore- 
shortened (B  in  Fig.  232) ;  and  lines  marking  its  horizontal  details 
(as  AB,  CD,  and  EF)  will  vanish,  as  would  any  parallel'horizontal 
lines.  If  now  we  move  this  object  toward  a  vertical  surface, 
placed  so  that,  vieived  from  above  it  mahes  right  angles  ivith  the  light 
rays  (as  shown  in  A,  Fig.  233),  the  shadoiv  on  the  vertical  surface 

148 


SHADOWS 


A.    JnowiNG- 
VERTICAL  Plane. 

ATUNEQOAU   ANGLES 
TO    5MAD0\A/   CENTEE 


B    Pei?spe:ctive   of  above 
—  shadow   distorted 

Fig.  234 


also  ivill   he  actuaUy  symmetrical;    though    it   may   appear  fore- 
shortened (B  in  Fig.  233),  as  in  this  case. 
Now  if  the  receiving  surface  be 

turned  so  it  is  not  at  right  angles 

(viewed   from    above)    with   the 

light  (Fig.  234),  we  get  what  we 

observed  in  Fig.  228,  —  an  actu- 
ally one-sided  shadow.    The  reason 

for  this  distortion  is  made  clear 

from  the  plan  in  Fig.  234.     The 

descending  light-ray  from  y  has 

farther  to  travel  before  striking 

the   receiving  surface,  hence  its 

shadow,    y,    is    lower    than    the 

shadow  from  z.     Such  variations 

of  original  shapes  are  of  the  same 

nature  in  producing  beauty  (and 

consequent  enjoyment)  as  theme 

variations  in  music.     Thus  the  bottle  with  shoulders  (Fig.  235) 

acquires  a  charm  from  the  proximity  of  its  interestingly  altered 

shadow-self  which   it   cannot   have   alone. 

For  the  shadows  of  curves  vertical 
'  planes  through  several  points  (as  x,  y, 
and  z)  are  taken  and  the  curve  then 
sketched  freehand.  The  use  of  this 
method  is  also  shown  in  drawing  the 
horizontal  hat-pin  in  Fig.  227  and  the 
the  vase  shadow  in  Fig.  228.  But  as 
soon  as  the  underlying  truths  are 
^^^-  ^^"^  clearly  understood,  the  actual  taking 

of  points  is  seldom  needed. 

So   far  the   shadows   have   fallen   on   flat   surfaces,  but   the 

shadow  of  the  vertical  box  edge  in  Fig.  228  falls  partly  on  the 

curved  book  back ;   and  on  this  it  forms  a  vertical  curve  —  that  is, 

with  its  ends  in  a  vertical  line.     The  curve  is  sketched  by  the 

149 


FREEHAND    PERSPECTIVE 


eye,  though  it  could  be  constructed  by  points.     In  Fig.  229  the 
shadow  on  the   book  back  is  cast  by  a  horizontal  edge  and  is 

therefore  an  oblique  curve.  In  this 
case  its  upper  point  (M)  is  found  by 
imagining  the  book  cover  continued 
until  it  cuts  the  back  of  the  box  in  a 
line  from  N  vanishing  with  the  long 
box  edges.  Where  the  shadow  line 
HI  cuts  this  line  (point  o)  will  be 
the  farther  end  of  the  shadow  line  on  the  book  cover.  This  line 
will  vanish  in  VPl,  and  where  it  cuts  the  upper  edge  of  the  book 
back  will  be  M,  the  upper  end  of  the  curve. 

The  book  in  Fig.  236  shows  the  use  of  points  when  the  edge 
casting  the  shadoiv  is  itself  oblique.    A  vertical  line  from  D  to  the 


SHADOWS 


A.   Plan,  J  MOWING 

T?At)IATINQ     RAVJ. 


(%P^  POINT  O,  THE 
V""^        CrAS    TLAMC 


edge  1-2,  and  a  line  from  that  point  (E)  to  C,  constructs  one 
vertical  plane.  The  light-ray  from  point  A  cuts  the  shadow- 
direction  of  AB  at  a.  The  shadow  of  CD  travels  from  C  through 
a  to  the  edge  1-2, 
and  from  there  to 
D. 

Shadows  on  a 
House.  —  The  truths 
thus  developed  ap- 
ply to  out-of-door 
work,  as  shown  in 
the  house  (Fig.  237). 
Here  one  new  con- 
dition is  met,  —  the 
shadow  of  the  ver- 
tical dormer  edge 
falls  on  the  oblique 
surface  of  the  roof, 
and  hence  has  an 
oblique  vanishing 
point.  This  vanish- 
ing point  is  easily 
found,  as  we  already 
have  two  points  in 
the  vanishing  trace 
of  theroof,  — OVPl 
and  VP2.  The  line 
containing  these 
points  is  the  vanishing  trace,  not  only  of  the  roof,  hid  of  the  infi- 
nite plane  containing  the  roof.  (It  therefore  can  be  drawn  as  long 
as  needed,  being  really  infinite  in  length.)  So  we  have  only  to 
draw  the  trace  from  OVPl  to  VP2,  and  mark  0VP3  on  it  verti- 
cally over  YP3,  exactly  as  we  marked  OVPl  over  VPl. 

The  shadow  of  the  bush  is  an  instance  of  the  ease  with  which 
shadow  laws  are  applied  to  natural  objects.  The  shadow  is  sketched 

151 


Perspective 

OF  ABOVE 


Fig.  238 


FREEHAND    PERSPECTIVE 


freehand ;  but  with  much  greater  certainty  for  knowing  that  its 
center  must  fall  on  the  ground  in  the  direction  of  VPS,  and  that  it 
can  extend  no  farther  than  its  meeting  with  the  ray  of  light,  HI. 
Shadows  from  a  Lamp.  —  The  radiating  rays  from  an  artificial 
light  can  all  be  contained  in  an  infinite  number  of  radiating  ver- 
tical planes  through  the  light  itself.  Some  of  these  radiating 
planes  are  seen  in  the  plan  (OA,  OB  and  others  in  Fig.  238). 

These  radiating 
planes  are  used 
instead  of  the 
parallel  vertical 
planes  previously 
explained.  Other- 
wise the  methods 
are  the  same  as 
with    light    from 


the  sun.  Thus  in 
the  shadow  of  the 
stool  the  light-ray 
from  O  through 
D  gives  d  where 
cut  by  a  line  on 
the  ground  di- 
rectly under  it 
^'^-  ~^^  (from    0    through 

E).  The  shadow  of  Gh  falls  on  the  floor  in  the  direction  of 
o-H  till  it  reaches  the  wall.  On  the  vertical  wall,  the  shadow 
of  the  vertical  GrH  is  also  vertical.  It  is  ended  by  the  light-ray 
from  O  through  Gr.  The  shadow  of  the  edge  GI  will  be  parallel 
to  it,  and  like  it  will  appear  as  a  vanishing  line  to  VPl.  The 
near  part  (d-j)  of  the  shadow  of  DI  will  be  parallel  to  DI,  and 
will  vanish  to  VPl  till  it  reaches  the  wall  at  J.  A  line  from  J 
to  I  completes  the  shadow  of  DI. 

Shadows   in   an   Interior.  —  These    are  partly  like  the   lamp 
shadows.     For  instance  the  shadow  on  the  couch  in  Fig.  239 

152 


SHADOWS 

extends  in  an  almost  opposite  direction  from  that  of  the  chair. 
These  shadows  are  produced  by  the  diffused  dayhght  radiating 
from  the  window.  On  the  other  hand  a  patch  of  sunlight  falling 
through  the  window  would  follow  the  laws  of  sunlight  generally. 
The  edges  a-l)  and  c-d  vanish  with  AB  and  CD,  while  points  a 
and  &  are  marked  by  the  meeting  of  light-rays  from  A  and  B 
with  shadows  of  verticals  from  A  and  B.  In  this  case  the  light 
comes  from  beyond  the  window,  hence  the  light-rays  recede  up, 
and  appear  to  converge  or  vanish  in  that  direction. 


153 


Chapter  XLV 
OUT-OF-DOORS  WORK 

A  LTHOUGH  the  same  perspective  principles  apply  to  out- 
/\      of-doors  work  the  conditions  of  the  study  vary,  and  some 
-A.   ■■^  cases  need  explanation. 

Vanishing  Points.  —  In  drawing  the  house  (Ch.  XXII),  we 
placed  ourselves  proportionately  in  relation  to  the  small  cube 
as  we  should  naturally  be  in  relation  to  the  real  house.  Thus 
the  sixteen  inches  of  distance  from  the  eye,  or  four  times  the 


51OE   VIEW  or  i-iousr     SMOwiMG   two  positions  op  eye 

Fig.  240 


height  of  the  cube,  was  equivalent  to  only  four  times  the  height 
of  a  twenty-foot  house,  or  eighty  feet  —  less  than  five  rods.  At 
this  short  distance  the  vanishing  of  the  lines  is  very  decided  ;  but 
at  a  half  mile  from  the  same  house,  those  lines  appear  nearly 
horizontal.  The  reason  for  it  is  seen  in  Fig.  240.  When  the  eye 
is  near  the  house  (at  x)  the  apparent  difference  in  length  between 
the  edges  AB  and  CD  is  greater  than  when  the  eye  is  at  y. 
This  is  shown  on  picture  plane  1  by  al)  and  ccl,  and  on  picture 
plane  2  by  a'l)  and  c'cT.  The  horizontal  edges  of  the  house 
as  seen  by  the  eye  from  x  would  therefore  vanish  more  steeply, 
causing  the  vanishing  point  to  fall  nearer,  as  shown  in  A,  Fig. 
241.  As  seen  from  y,  the  horizontal  edges  are  less  steep ;  there- 
fore in  B  the  vanishing  points  fall  much  farther  away. 

It  follows,  therefore,  that  the  greater  the  distance  of  the  eye 

154 


OUT-OF-DOORS     WORK 


from  an  object ,  the  farther  to  right  and  left  will  the  vanishing 
points  fall.  When  the  house  is  a  half  mile  away  they  fall  so  far 
to  left  and  right  that  its  horizontal  lines  appear  almost  level. 
Hence  the  beginner  in  landscape  work,  accustomed  only  to  near 
objects,  is  sometimes  puzzled,  because  distant  houses  seem  to 
have  no  perspective.     And  the  landscape  artist  who  has  "  no 


.aJ^^^.,, 


SHAPE   OF  HOUSE  AS 
5CEN    FROM     X 


B       SHAPE    OF  HOUSE   AS 

SEEN  reoM  Y, 


Fig.  241 


trouble  with  houses  "  in  the  distance  may  shrink  from  attempting 
them  in  the  foreground. 

Size  of  Objects  Seen.  —  The  image  formed  on  the  retina  of  the 
eye  is  always  exceedingly  small,  and  with  distant  objects  becomes 
microscopic.  All  mental  picturing  of  the  size  of  objects  pro- 
ceeds from  our  mental  knowledge  of  their  actual  dimensions. 
Size  judged  from  seeing  alone  can  be  but  a  matter  of  comparison. 
This  is  easily  proved  by  asking  two  persons  how  large  the  moon 
appears  to  them.  Here  we  have  an  object  whose  real  size  and 
distance  are  so  great  as  to  be  no  guide  in  comparison  with  other 
objects  and  it  will  probably  appear  of  a  different  size  to  each 
person.  There  is  consequently  no  such  thing  as  the  drawing 
of  objects  ''  the  size  they  appear."  Size  in  draiving  is  merely 
relative;  and  the  scale  on  which  a  drawing  is  made  is  ivholJy  a 
matter  of  choice.  We  may  choose  to  make  a  drawing  what  is 
termed  "  actual  size,"  but  this  means  that  we  regulate  its  size  by 
a  mental  knowledge  obtained  either  from  measuring  in  the  ordi- 
nary way,  or  by  putting  our  sketch  back  by  the  object  to  compare 
them  by  the  eye. 

The  absolute  size  of  objects  varies  so  much,  also,  that  unless 
the  picture  contains  something  the  size  of  which  is  well  known 

155 


FREEHAND    PERSPECTIVE 


and  but  little  variable,  we  cannot  be  sure  of  the  sizes  repre- 
sented. The  human  fig- 
ure serves  best  for  such 
a  standard,  but  some 
objects  always  adjusted 
to  the  human  figure  in 
size,  as  steps,  and  often 
doors,  will  answer  in  its 
place. 

TJie  size  of  objects  ac- 
cording to  their  distance 
iiito  the  picture  is  impor- 
tant in  out-of-doors  work 
also.  Here  the  indis- 
pensable picture  plane 
becomes  again  useful. 
In  Fig.  242,  for  instance, 
the  gondolier  must  not 
be  too  large  for  the 
buildings.  Lines  drawn 
from  his  head  and  a 
Fig.  242  poiut  ou  the  watcr  di- 

rectly under  it  to  the  eye  level  will  contain  between  them  his 
heisfht  above  the  water 


all  the  way  to  their  van- 
ishing point.  If  we 
wish  to  know,  for  in- 
stance, whether  the  door 
on  the  left  is  large 
enough,  we  have  only 
to  draw  horizontal  lines 
from  its  top  and  from  a 
point  directly  under  it  on 
the  plane  of  the  water  ^^^-  -^^ 

continued.    Where  this  water  line  would  cut  the  ^ater  line  from 

156 


OUT-OF-DOORS     WORK 


the  figure  to  the  vanishing  point  a  vertical  line  is  erected,  on 
which  the  two  heights  can  be  compared. 

Reflections.  —  If  a  mirror  be 
laid  on  a  table,  and  a  cup  placed 
on  it  (Fig.  243),  the  reflection 
will  appear  precisely  like  the 
cup  reversed,  with  its  bottom 
resting  against  the  bottom  of 
the  real  cup.  The  reflection 
will  not  present  to  the  eye  the 
same  shape  as  the  real  cup, 
for  besides  being  reversed  it  is 
farther  below  the  eye,  making 
its  inside  invisible  while  its 
base  is  covered  by  that  of  the 
real  cup.  We  can  also  see 
farther  around  on  its  flaring 


-7-x 


A.   Perspective- 

SHOVl/lNG   STAKE 
FORESHORTENED 


B.  Side  view  of  stake  -  showing  how 
its  foreshortening-    occurs. 

Fig.  244 

surface,  because  its  decrease  of  diameter  is  toward  the  eye  level, 
while  in  the  actual  object  it  is  away  from  the  eye  level. 

Now  since  it  is  like  the  cup  reversed  we  see  that  any  point 
(as  A)  in  the  cup,  must  be  reflected  directly  under  itself.  So  a 
stake,  thrust  into  a  pool  of  still  water  (Fig.  244),  will  produce 
a  reflection  like  itself  reversed ;  ^  and  each  point  in  the  reflec- 
tion will  be  directly  under  the  same  point  in  the  real  stake. 

^  Tn  this  case  appearing  longer  than  the  real  stake,  as  explained  a  few  pages  later. 

157 


FREEHAND    PERSPECTIVE 


It  is  therefore  evident  that  in  case  of  reflections  on  a  horizon- 
tal surface  J  the  image  formed  must  he  vertically  under  the  reality. 

Consequently,  as  long  as  the 
reflecting  surface  remains 
horizontal,  reflections  on  it 
cannot  be  thrown  to  one 
side,  but  must  be  shown  di- 
rectly under  the  real  objects, 
even  if  the  reflecting  surface 
be  broken  (as  in  Fig.  242). 
And  if  in  drawing  reflections 
we  represent  them  out  of 
the  vertical  (Fig.  245)  the 
reflecting  surface  (in  this 
case  the  water)  appears  to  be 
sloping,  like  rapids  in  a  river. 
We  may  therefore  take  as 
our  rule  that  reflections  are 
invariably  like  the  reflected 
object  reversed  on  the  reflecting 
plane. 

When    the   Object    is    Sep- 
arated   from    the    Reflecting 
Surface.  —  In  Fig.  246  the  bungalow  is  separated  from  the  reflecting 


WRONG.  Reflections  not  ver- 
tically UNDER  OBJECTS   RE- 
FLECTED. WATER  APPEARS 

sloping- not  leueu. 

Fig.  245 


Ihls  ctistonce   (E  f)    further  into  the 
picture  thar\  tb>e  bocJ'Wou3e., 


Fig.  216 
158 


OUT-OF-DOORS    WORK 


surface  (the  water)  by  a  high  bank.  But  by  using  a  water  line 
of  the  boathouse  (which  stands  parallel  to  it  and  directly  on  the 
water),  the  points  (A,  B,  and  C)  where  the  bungalow  edges 
continued  would  strike  the  plane  of  the  water  can  be  closely 
approximated.  From  there  the  points  (a,  b,  c,  and  d)  for  the 
reflection  of  the  bungalow  are  measured  vertically. 

Even  if  we  had  not  the  boathouse  to  give  parallel  vanishing  lines 
on  the  water,  the  necessary  points  (A,  B,  and  C)  could  be  esti- 
mated with  sufficient  accuracy  after  a  little  experience.  The  main 
thing  to  remember  is  that  it  is  on  the  reflecting  surface  or  07i 
its  plane  continued  that  the  object  is  reversed  in  its  reflection. 

Reflections  on  Vertical  Surfaces.  —  With  reflections  on  vertical 
surfaces  the  problem  is  very  simple.  In  Fig.  247  the  box  appears 
reversed  as  far  back 
of  the  mirror  surface 
(the  thickness  of  its 
frame)  as  it  actually 
stands  in  front  of  it. 

Length  of  the  Re- 
flection. —  The  verti- 
cal length  of  the  re- 
flection, while  the 
reflecting  surface  is 
unbroken  (as  in  Fig. 
243)  is  actually  the 
same  as  that  of  the  real  subject.  This  does  not  mean  that  the 
reflection  will  always  appear  of  the  same  vertical  length  as  the 
object,  as  that  depends  on  its  position  and  on  the  location  of 
the  point  from  which  it  is  viewed.  In  Fig.  244  the  stake  is 
seen  from  a  higher  point  and  leans  toward  the  beholder.  It  is 
consequently  seen  foreshortened,  as  the  roundness  of  its  top 
indicates.  The  reflection,  being  reversed,  appears  practically  in 
its  true  length.  A  point  (x)  on  the  surface  of  the  water  directly 
under  the  top,  appears  lower  than  where  the  stake  enters  the 
water,  because  nearer  the  eye.     Cases  like  the  familiar  "  silver 

159 


Fig.  247 


FREEHAND    PERSPECTIVE 

path "  of  the  moon  in  rippUng  water,  or  like  Fig.  242,  where 
the  reflections  of  upright  objects  appear  lengthened  vertically 
as  well  as  broken,  are  caused  by  the  many  curved  surfaces  of  the 
waves  on  which  successive  bits  of  the  reflection  fall. 

Use  of  the  Finder.  —  Nowhere  will  the  finder  (Ch.  YIII)  be 
of  more  use  than  in  out-of-doors  work.  The  difference  in  distance 
between  the  near  and  far  objects  in  a  landscape  is  so  great, 
that  the  beginner  finds  it  hard  to  realize  how  much  difference 
he  must  make  in  size.  The  finder  serves  as  a  measuring  unit 
for  these  differences,  besides  being  invaluable  as  an  aid  in 
selection. 


160 


SOLUTIONS  OF  PROBLEMS 


T 


CHAPTER  XI  — PAGE  34 

THE   CYLINDER  CONE  AND   BALL 

HE  Cone.  —  After  the  cylinder  is  drawn,  the  base  of  the  cone  is 
next  placed.  This  is  actually  a  circle,  and  of  the  same  size  as 
the  cylinder  base.     Its  position  will  be  clear  from  the  plan  (Fig. 


249,  p.  34).  In  per- 
spective it  is  best 
placed  by  its  true 
center.  If  the  cone 
were  moved  on  the 
ground  around  and 
touching  the  cylin- 
der, this  center 
would  describe  a 
circle,  twice  the 
diameter  of  the 
cylinder  base,  and 
equally  distant  at 
every  point  from 
the  cylinder.  This 
circle  is  sketched  in 
perspective  (Fig. 
250)  as  an  ellipse 
(see  Chs.  IV  and 
XX).  The  true 
center  for  the  base 
of  the  cone  is  placed 
on  this  ellipse  (at 
0).  From  this 
true  center  a  ver- 

11 


:^^^s^'r:^  *:rjtoL«fSs2t!i3»p;«;2s^«*sss*i:.n-ts: 


►•«-^«Mt»ii^>Sti^jifo:&2iS»itjK«a<ii;S>;iC-,i?ia««iSi;^^ 


ZXSajJUSVl.flt.gX.is:^., 


Fig.  248 


161 


FREEHAND    PERSPECTIVE 


Fig.  249 


tical  line  of  indefinite  length  may  be  erected,  on  which  to  set  off  the 
axis  of  the  cone.  Its  height,  being  actually  the  same  as  that  of 
the  cylinder,  will  appear  slightly  greater  because  nearer  the  eye.  At  the 
same  time  its  apex  (F),  being  nearer,  cannot  appear 
quite  so  high  on  the  paper  as  even  the  nearest  edge 
of  the  cylinder  top.^  Through  its  lower  end  (0)  the 
real  diameter  of  its  circular  base  passes.  Being  at 
the  same  distance  into  the  picture  as  the  axis,  and 
like  it  parallel  with  the  picture  plane,  it  appears 
in  its  true  proportion  to  the  axis  (one  half).  It  is 
therefore  so  set  off,  equally  on  each  side  of  0, 
giving  AB.  If  right,  this  true  diameter  should  measure  a  little  greater 
than  the  diameter  of  the  cylinder.  The  base  of  the  cone,  though  actually 
of  the  same  size  as  the  cylinder,  will  ajyjiear  both 
a  little  larger  and  a  little  rounder,  because  nearer. 
The  short  diameter  of  this  base  (CD)  is  therefore 
set  off  greater  than  that  of  the  cylinder  base, 
remembering  that  as  0  is  the  real  center,  DO 
must  be  larger  than  CO.  The  long  diameter  (the 
light  line  in  front  of  AB)  is  next  drawn,  exactly 
in  the  middle  of  CD  and  a  very  little  longer  than 
AB.     The  ellipse  is  then  sketched   through   the 

four  points  A,  B,  C,  and  D,  taking  care  to  have  it  touch  the  base 
of  the  cylinder,  and  to  make  the  greatest  length  not  on  AB,  but  on 
the  long  diameter.  The  cone  is  completed  by  drawing  its  sides  from 
the  apex  tangentially  to  this  elliptical  base. 

The  Ball.  —  If  the  ball  be  rolled  about  and  touching  the  cylinder  it 
will  follow  the  same  path  as  the  center  of  the  cone  base,  so  that  its 
resting  point  will  always  be  somewhere  in  the  ellipse  representing  that 
path  (its  center  being  always  vertically  above  the  resting  point).  We 
should  therefore  mark  some  point  in  the  large  ellipse  (in  this  case  x") 
for  the  resting  point  of  the  ball.     If  we  stoop  to  bring  the  eye  nearly 

^  r>eiiig  actually  of  the  same  height,  they  lie  in  the  same  horizontal  plane.  This  plane, 
being  below  the  level  of  the  eye,  appears  to  recede  upward,  as  the  table  does.  This  will  be 
better  understood  if  a  sheet  of  paper  is  laid  on  the  tops  of  the  two  objects,  when  it  can  be  seen 
that  it  appears  to  recede  upward.     The  following  chapter  will  further  illustrate  this  truth. 

162 


Fig.  250 


SOLUTIONS    OF    PROBLEMS 

to  the  table  level,  and  look  at  the  ball,  we  shall  see  it  resting  on  this 
spot.  But  if  we  return  to  the  point  from  which  the  group  was  to  be 
viewed,  we  shall  find  this  point  hidden  by  the  projecting  mass  of  the 
ball.  The  circle  which  represents  the  boundary  of  the  ball  is  therefore 
drawn  with  its  lower  edge  a  little  below  x,  and  its  center  vertically  over 
that  point.     Being  nearer  the  eye,  its  diameter  is  greater  than  AB. 


CHAPTER  XV 

THE  CYLINDER  AND  THE  RECTANGULAR  BLOCK 

PAGE  48 


The  Block.  —  This  should  be  drawn  first 
parallel  with  the  pic- 
ture plane,  it  will 
appear  in  its  true 
shape,  and  the  block 
ends  will  vanish  in 
VPl  like  the  book 
ends  in  Chapter  XH. 
In  setting  off  the 
apparent  width  of 
the  top,  we  remem- 
ber that  it  is  actually 
narrower  in  propor- 
tion than  the  book 
cover. 

The  Cylinder.  — 
The  cylinder  rests 
against  this  block 
(side  view,  Fig.  252), 
so  we  can  measure 
the  height  of  its  back 
(AB,  Fig.  253)  actu- 
ally, making  it  twice 
the  height  of  the  block 


Since  its  front  face  is 


Fig.  251 


163 


FREEHAND    PERSPECTIVE 


51DE  VIEW  or  GCOUP. 

Fig.  252 


front.  The  lower  base  is  actually  the  same  in  width  as  the  block,  but  be- 
ing nearer  the  eye  it  will  appear  larger.  Just  how  much  can  be  easily 
determined.  Draw  the  invisible  lines  of  the  block  (the  dotted  lines  in 
Fig.  253),  and  carry  the  di- 
agonal of  one  half  (x)  for- 
ward in  a  line  of  indefinite 
length.  Cut  this  by  line  y 
of  the  invisible  edge  of  the 
block.  From  the  point  (C) 
so  found  draw  line  s  to  the 
right,  to  cut  another  invisi- 
ble edge  continued.  This  constructs  another 
rectangle  the  actual  size  of  the  side  of  the  block, 
but  nearer,  hence  appearing  larger.*  The  middle 
of  the  front  of  this  rectangle  (point  D)  will  be 
the  front  of  the  base  of  the  cylinder.  Its  back 
will  be  A,  and  its  long  diameter,  EF,  can  be  set  off  on  a  line  sketched 
half  way  between  this  front  and  back,  marking  E  and  F  half  way 
between  AD  and  the  ends  of  the  construction  rectangle.  Through  these 
four  points  the  bottom  ellipse  is  drawn. 

For  the  top  ellipse  a  similar  rectangle  can  be  constructed  directly 
above  it.  This  will  give  a  much  more  foreshortened  ellipse,  as  would  be 
expected.  The  back  and  front  of  the  middle  ellipse  are  drawn  as 
directed  in  Chapter  IV. 


Fig.  253 


CHAPTER   XXVI  — PAGE  91 

THE  SQUARE  FRAME  LEANING  ON  A  REC- 
TANGULAR BLOCK 

The  Rectangular  Block.  —  This  solid  is  drawn  in  the  same  manner 
as  the  book  similarly  placed.  Recall  that  it  is  equal  to  two  cubes,  as 
shown  in  Fig.  255,  where  a  diagonal  (AB)  of  the  horizontal  top  face 


1  This    use    of  the  diagonal  for  measuring  will  be    found  in    Chapters   XXII,    XXV,   and 


others. 


164 


SOLUTIONS    OF    PROBLEMS 


of  the  first  cube  is  carried    to  its  vanishing  point  VPS,  from  which  a 
perspectively  parallel  line  is  drawn  back  to  C,  cutting  the  back  edge 


Fig.  254 

in  D,  which  becomes    the    farthest  corner 
of  the  second  cube,  and  of  the  whole  block.    v${^, 
A  line  through  D,  vanishing  in  VPl  gives 
the  corner  E. 

The  Frame.  —  For  this  the  loiver  face 
only  may  be  first  considered.  This  face 
rests  on  AE  in  points  one  eighth  of  AE 
from  either  end,  which  can  be  easily  found 
by  the  method  used  on  page  79  for  the  steps.  (The  vertical  edge 
AH  is  divided  into  eighths,  and  lines  from  points  1  and  2  vanished 
to  VP2.  Where  they  cross  the  diagonal  Al  verticals  are  erected, 
marking  points  F  and   G.) 

Now  by  placing  a  card  against  one  of  the  side  thicknesses  of  the 

165 


PERSPECTIVE    VIEW. 
THE  LI  IVES  ABAND  CD 

are  actually  parallel 
am0van16hin  vp3.  _.  / 

Fig.  255 


FREEHAND    PERSPECTIVE 


Fig.  256 


frame. 


frame  (Fig.  255)  we  readily  see  that  these  thicknesses  are  vertical,  and 
that   their   vertical    planes    (one  of  which    is   indicated    by    the  card) 

are  parallel  to  the  block  ends,  and  cut 
the  block  side  A  E  I  H  in  vertical  lines 
(FJ  and  GK,  already  drawn),  and  also  cut 
the  ground  in  horizontal  lines  through  J 
and  K,  parallel  to  the  horizontal  lines  of 
the  block  ends.  In  these  horizontal  ground 
lines  rest  the  lower  corners  of  the  leaning 
We  therefore  vanish  the  lines  through  J  and  K  to  VPl, 
continuing  them  forward  indefinitely.  The  distance  JL  (Fig.  257) 
is  actually  equal  to  half  the  block  width ;  that  is,  to  J3,  and  can 
be  easily  measured  by  a  line  through  J  parallel  to  H3  (vanishing 
with  it  to  VP4).  From  where  this  line  meets  the  continuation  of  the 
edge  HN  (in  point  4)  a  line  parallel  to  HI  (vanishing  in  VP2)  gives 
the  corners  L  and  M,  and  the  ground  edge  of  the  frame. 

The  leaning  edges  of  the  square  are  drawn  from  L  and  M  through 
F  and  G,  and  will  be  found  to  vanish  in  OVPl.  They  vanish  a  little  less 
(have  a  more  distant  vanishing  point)  than  the  horizontal  edge  LM,  and 
should  therefore  be  made  slightly  longer,  but  shorter  than  if  standing 
erect  from  L.  We  can  check  our  estimate  by  comparison  with  LO, 
the  height  of  the  frame  at  that  point  (EH).  To  obtain  this,  the  height 
of  the  block  at  J  is  carried  for- 
ward to  P  on  the  vertical  (by 
lines  vanisliing  in  VPl)  and  one 
half  of  LP  added  at  the  top, 
making  LO. 

The  Thickness  Edges. — 
These  edges  must  vanish  sharp- 
ly (or  have  a  near  vanishing 
point)  because  the  edges  at 
right  angles  to  them  (the  ones 
to  OVPl)  are  foreshortened  but 
little.  Hence  0VP2  is  placed 
but  little  below  the  group.     To  this  point  these  short  edges  are  drawn, 

IGG 


-X      -TiVPi: —  -^ 


T^^HSi-* 


Fig.  257 


Fig.  258 


SOLUTIONS    OF    PROBLEMS 

carrying  them  forward  of  the  corners  indefinitely  for  a  short  distance. 

The  foreshortening  of  these  edges  (actually  one  sixth  of  the  long  edges  of 

the  square)  will  be  much  greater  than  that  of  the 

long  edges ;  and  a  corner  is  marked  accordingly 

at  Q.     From  this  corner  a  horizontal   edge  van- 
ishes to  VP2,  and  a  long  oblique  one  to  OVPl, 

giving  respectively  R  and  S  where  each  crosses 

a  thickness  edge.     From  S  the  other  horizontal 

edge  vanishes  to  VP2  and  from  R  the  other  long  oblique  one  to  OVPl, 

completing  the  square. 

The  Inner  Square.  —  This  is  drawn  as  on  page  86. 
The  frame  width  can  be  measured  perspectively  on 
LT  by  continuing  the  edge  TS  to  cut  the  con- 
tinuation of  LO  in  point  6,  dividing  L5  into  six 
actually   equal   parts  and  vanishing  lines  to  0VP2 

THl^LiN^E'l'^HreHll°na  as?    froui  the  first  and  last  of  these  division  points,  eriv- 

VANISH  IN  bvP*  '^RE  HERE  SEEN  ■«•  ^    <=> 

To«.AeTUAuuy.«.RALLx..    -^^^  Q  ^^^  ^^  ^j^^  ^^^^^^  nccded. 

Fig.  259 


CHAPTER   XXXI 
PAGE  102 

THE  CLOCK 

The  dial  of  the  clock  is  expressed 
by  the  methods  used  for  the  circu- 
lar frame  in  the  square  frame  (Ch. 
XIX).  After  the  rectangular  por- 
tion of  the  clock  has  been  sketched, 
an  enclosing  square  for  the  dial  (AB 
CD)  is  drawn  and  the  clock  axis 
vanished  from  the  actual  center  of 
this  square.  The  axis  is  continued 
forward  indefinitely  (upward  to  left) 
to  give  the  direction  of  the  short  di- 
ameter (which  always  apjjears  to  lie  in  line  with  the  axis, 
actually  at  right  angles  to  it  —  p.  97,  Fig.  160) .      For  the  ends  (G 

1G7 


Fig.  260 


though 
andH) 


FREEHAND    PERSPECTIVE 


of  the  short  diameter,  slightly  curved  lines  are  sketched  up  to  the  right 
from  E  and  in  reverse  direction  from  F,  giving  the  front  and  back  of  the 
ellipse.  The  long  diameter  is  then  sketched,  of  an  indefinite  length, 
midway  between  G  and  H  and  at  right  angles  to  the  axis.  It  is  then  cut 
by  curves  from  I  to  the  right,  and  from  J  to  the  left,  being  careful  to 
keep  K  and  L  equally  distant  from  the  axis.  The  remainder  of  the  el- 
lipse may  then  be  drawn,  correcting  the  curves  as  needed ;  and  even 
moving  K  and  L  if  necessary  for  symmetry,  but  not  changing  points  E, 
F,  I  or  J.  The  inner  ellipse  is  drawn  in  the  same  way,  as  explained  on 
page  98. 

CHAPTER  XL -PAGE  131 

THE  TRIANGULAR  PRISM  AND  FRAME 

The  Triangular  Frame. 

—  This  solid  is  easily  con- 
structed by  the  use  of  the 
cube  (see  dotted  lines). 
The  length  of  AC  is 
found  by  the  diagonal 
(ED)  of  one  side  of  this 
imaginary  cube,  DF  being 
made  equal  to  DG  (p. 
165).  The  steps  in  Chap- 
ter XXII  illustrate  this 
method.  The  end  view 
shows  how  the  triangle 
is  related  in  shape  to  the 
square  face  of  the  cube. 
Its  vertical  center  line  is 
located  by  the  diagonals 
of  the  square,  and  the 
height  of  its  apex  is  meas- 
ured to  X  on  the  near  verti- 
cal edge  of  the  cube  (AE). 
The  Triangular  Frame.  —   I 

This  is  also  readily  drawn  Fig.  261 

168 


SOLUTIONS    OF    PROBLEMS 

by  the  help  of  one  face  of  the  cube.  After  the  triangular  outline  is 
sketched,  the  height  of  one  lower  bar  (one  sixth  of  the  height  of  the  cube) 
is  marked  upward  from  C,  giving  point  E,  and  the  lower  edge  of  the 
inner  triangle  is  vanished  through  E.  Where  BF,  drawn  so  as  to  divide 
AD  perspectively,  crosses  this  lower  edge  is  a  corner  of  the  inner  triangle. 


Fig.  262 


Fig.  263 


Through  this  corner  (G)  another  edge  of  the  inner  triangle  vanishes  to 
OVPl.  The  other  edge  is  drawn  toward  0VP2.  The  thicknesses  are 
found  as  in  the  square  frame  (Ch.  XXIV). 


169 


INDEX 


Aims  of  perspective,  drawing  objects  as 
they  appear,  140 
learning  to  see,  xii 

to  acquire  artistic  judgment,  80,  83, 
120 
Apparent  size  of  objects,  according  to  dis- 
tance, xi,  156 
in  out-of-door  work,  155 
in  relation  to  other  parts  of  picture, 

108 
relative  only,  6,  155 
Arch,  errors  in  drawing,  99 

pointed  and  other  forms,  104 
round,  99,  103-104 
Artistic  judgment,  80,  83,  120 
Artistic  rendering,  book,  59 
buildings,  83 
glass,  25,  28,  30 
rose  jar,  17 


Background,  subordination  of  objects  in, 
fan,  33 

leaning  bowl,  26 

plate,  30 
Bases  of  cylindrical  objects,  see  Foot 

always  partly  visible,  47 

location  on  horizontal  surfaces,  29,  30, 
32,  161-162 
Baskets,  90 
Benefit  of  perspective  study,  xi 

acquiring  of  artistic  judgment,  80,  83, 
120 

learning  to  see  correctly,  xii 
Book,  artistic  rendering  of,  59 

at  angles  to  picture  plane,  58-60 

back  of,  44 

clasps  of,  44 

cover  thickness,  44 

in  two  positions,  43,  44 

margins,  40,  41 

projection  of  covers,  44 

use  of  pencils  to  show  convergence  of 
lines,  37 

use  of  strings,  38,  39 

vertical  edges,  60 


with  back  parallel  to  face,  38-42 
with  cylindrical  object,  46-47 

Books,  two,  at  different  angles  to  the  pic- 
ture plane,  61,  62 
with  a  cylindrical  object,  67,  68 

Boundary,  movable,  16,  17,  21 
tangential  to  ellipses,  13,  17 

Buildings,  camera  distortions  in,  139,  142 
few  vanishing  points  for,  84 
from  photograph  or  print,  81 
house,  the,  69-80;  see  House 
round  arch,  the,  99,  103 
round  window,  the,  100,  101 
spire  or  tower,  90 
type  forms  useful  in,  85,  88 


Camera  distortions,  139,  142 
Carrying  lines  "around  a  corner,"  86,  130 
Central  direction  of  seeing,  alluded  to,  10, 
140 
explained,  6 

moves  with  changed  picture  center,  46 
Chair,  the  study  of,  118-120 
Circle,  actual  center  of,  63;  see  Ellipse 
concentric  circles,  14,  63-66 
location  of  its  center  in  ellipse,  15,  65 
only  position  in  which  seen  as  circle, 

xii 
see  obliquely,  9 
Circular  frame  within  square  frame,  96-99 
application  of  its  principles,  99;    see 
Cylindrical  objects  not  vertical 
Clock,  102,  167 
Color,  in  buildings,  83 
of  book,  59 
on  rose  jar,  17 
Composition,  cylinder  and  cylindrical  ob- 
ject,  12 
cylindrical  objects  grouped,  26;   with 

books,  68 
in  selecting  from  interior,   112,   115; 
from  photograph  of  building,  81 
Concentric  circles,  14 

with  square,  63-66 
Cone  model,  18 


171 


INDEX 


Cone  principle,  19 
Cover  of  teapot,  28 
Cream  jug,  foot,  22 

handle,  20 

ornament,  23 

spout,  22 

study  of,  20-23 
Cube,  at  45°  with  picture  plane,  53,  56 

at  30°  and  60°  with  picture  plane,  50 

making  the  drawing,  54 

order  of  drawing  edges,  54 

proportions  used  in  estimating  other 
objects,  51,  86,  129,  164-165, 
167-168 

recession  of  horizontal  surfaces  to  eye 
level,  51 

relation  of  foreshortening  to  vanish- 
ing of  edges,  50,  51 

study  of,  48-52 

taking  direction  of  edges  with  pencil, 
55 

tests  of  vanishing  lines  by  string  and 
by  eye,  56;  on  blackboard,  57 
Cylinder,  errors  in  ellipses  of,  14 

hollow,  the,  14,  15 

inner  cylinder,  14 

models  for,  8,  14 

perspectively  equal  divisions,  15 

position  of  model,  13 

roundness  of  ellipses,  14-16 

sides  tangential  to  ellipses,  13 

study  of,  12-15 

symmetry  of  ellipses,  15 

true  diameter  of  circle,  15 
Cylindrical  objects  grouped,  26-28 
Cylindrical  objects  not  vertical,  92-94 

application  of  principle,  94 

other  examples,  button  on  cord,  19; 
circular  frame,  96-99;  clock, 
102;  flower  pots,  95;  leaning 
bowl,  27;  luncheon  carrier,  32; 
round  arches,  99,  103-104;  round 
window,  100,  101 

symmetry  of  appearance,  93-94 

test  for  drawing  of,  94,  95 
Cylindrical  objects  with  fruit,  29,  30 
Cylindrical  picture  plane,  140-141 


Desk,  179 

Diagonals,  use  for  measuring,  concentric 
circles,  65,  66 
door  in  room,  108 


square  frame,  86 
square  plinth,  89 
Drawing    from    a    description,    xii; 
Problems 


see 


Ears  of  teapot,  28 

Ellipse,  at  right  angles  to  axis  in  cylindrical 
objects,  93-94 

common  errors  in,  14 

diameters  of,  9,  15 

drawn  entire  first,  14,  15 

from  concentric  circles,  14,  63-66 

measurement  on  its  diameters,  14,  15 

position  of  hand  in  drawing,  11 

practicing,  10,  11 

roundness   according  to  position,   9, 
10,  13-15 

study  of,  8-11 

symmetry,  9,  15 

tangential    to    boundary    lines,     13, 
15,  17 

test  of  shape,  9 

true  diameter  of  circle,  15 

varying  curvature  of  boundary  line,  9 
Exceptions  to  the  use  of  the  flat  picture 
plane,  139-142 

cylindrical  picture  plane,  140-141 

spherical  picture  plane,  141-142 
Eye  level,  explained,  52 

finding,  39,  40 

importance,  40 

way  of  using,  39 

Fan,  33 

Finder,  26,  67,  160 
Flower  pots,  95 

Foot  of  cylindrical  objects,  at  least  partly 
visible,  47 

of  cream  jug,  22,  23 

of  rose  jar,  16,  17 
Foreshortening,  xi,  10 
Foundation  truths  of  perspective,  two,  xi 
Freehand  sketching  defined,  xii 
Freehand  work  entirely,  3 
Fruit   grouped   with   cylindrical   objects, 

29-30 
Furniture,  chair,  118 

clock,  102,  167 
( desk,  179 

good  examples,  120 

rocking  chair,  118 

proportioning  to  other  objects,  108 


172 


INDEX 


Geometric  solids,  omitted  at  the  teacher's 

discretion,  85,  note 
Geometric   measurements,    obtained   per- 

spectively,  66,  108 
Glass,  bowl,  25 
pitcher,  30 

Handle,  cream  jug,  20-22 
Hexagonal  plinth,   application  of  study, 
126-127 

test  for,  124-125 

two  positions,  121-125 
Hexagonal  prism  and  frame,  128-130 

estimating  length  of  prism,  129 
Horizontal  surfaces  foreshortened,  37,  40, 

41 
Horizontal  surfaces  recede  to  eye  level,  51 
Horizontal  vanishing  edges,  37-39,  50 
House,  69-80 

chimney,  76 

dormer  window,  79 

eaves  projections,  74 

"L"part,  75 

model,  69 

porch,  75 

roof,  70-73 

steps,  78 

windows  and  doors,  75-76 
How  much  to  include  in  the  picture,  112 

Interiors,  at  angles  to  picture  plane,  110- 

113 
ceiling,  little  or  none  shown,  115 
door  in  an  interior,  107 
from  memory,  106  and  note 
further  studies  of,  114-116 
lines  must  not  point  to  corners,  109, 

116 
parallel    to    picture   plane,    105-109, 

132,  133 
picture  on  wall,  106 
relation   of   subject-space   to   picture 

plane.  111 
selection  of  subject-space,  110 
stool,  108 
with  tiled  floor,  126-127 

Knob  of  teapot  cover,  28 

Lamp  shade,  18,  19 

Line,  directions  for  drawing,  1,  2 

expressive,  17 

texture  of,  1 


Lines  of  the  picture  must  not  make  equal 
angles,  115 
nor  run  to  corners  of  margin,  109,  116 

Margin  of  picture,  1,  13 

cutting  the  group,  27,  illus.,  28,  30 
moving,  to  improve  picture,  116 
partial,  116 
Margins,  of  the  book,  40,  41 
Materials,  pencil  and  paper,  1 

models,  2;  see  Models 
Measuring,  by  the  diagonal,  65,  86,  89, 
107-108 
distance  into  the  picture,  65-66 
height  within  the  picture,  90,  108 
only  relative,  6,  66 
Measurements  obtained  geometrically  can 
be  so  obtained  perspectively,  66, 
107-108,  122-123,  164-165,  167- 
168 
Mechanical  perspective,  value  alluded  to, 
140 
errors  of,  139 
correction  of,  140 
limitations  of,  142 
Memory  work,  conditions  of,  31 
from  interiors,  106 
group  of  objects  from,  31-32 
less  laborious,  note,  106 
necessity  for,  xii 
specially  advised,  43,  53,  58,  67 
Methods,  their  subsequent  use  in  practical 

work,  xii,  101 
Models,  in  general,  2 

making,   cone,    18,    19;    cylinder,   8; 
cube,  49;   rectangular  block,  48; 
hexagonal  plinth,  121;  triangular 
prism,  131 
position  for  drawing,  2 


Oblique  vanishing  lines,  chair,  119 

dormer,  80 

hexagonal  frame,  129,  130 

Hght  rays,  147 

roof  of  house,  72 

shadow  on  roof,  151 
Obstacle  to  mastery  of  perspective,  xii 
Ornament,  constructive  principles  of,  23 

on  Japanese  luncheon  carrier,  32,  33 

rendering  of,  17 

use  in  composition,  13 
Out-of-doors  work,  154-160 


173 


INDEX 


greater  distance  of  vanishing  points, 

154-155 
reflections,  156-160;  see  Reflections 
size  of  objects,  155 

Parallel   retreating    lines,    convergence 

of,  40,  50 
Parallel  retreating  horizontal  lines,  meet- 
ing at  eye  level,  50,  93 
Parallel  perspective,  bureau,  134 
interiors,  105-109,  132-133 
street,  134,  136-137 
term  "parallel"  unsatisfactory,   135, 
note 
Pencil  measurement,  difficulties  of,  6 
essential  requirements  for,  6 
gives  relative  size  only,  6 
of  a  door,  7 
of  the  book,  39,  40 
of  the  cube,  55,  56 
of  the  ellipse,  13 
on  a  window-pane,  5 
on  the  picture  plane,  7 
study  of,  4-7 
Picture  plane,  different  for  each  picture, 
46,  113 
exceptions  to  use  of  flat  picture  planes, 

139-142 
may  be  assumed  at  any  distance,  6 
method  of  using,  7 
position  relative  to  central  direction 

of  seeing,  6 
relation  to  group  of  objects,  46 
relation    to    subject   in    drawing    in- 
terior, 110-112 
study  of,  4-7 
Position,  for  drawing,  1 

of  hand  for  ellipses,  1 1 ;  for  lines,  1-2 
of  models,  2 
Practical  use  of  methods,  xii,  lOl 
Practice,  of  ellipses,  11 

of  lines,  1,  2 
Principles    of    perspective,    two    founda- 
tion, xi 
Problems,  clock,  102 

conditions,  general,  34;  special,  102 
cylinder,  cone  and  ball,  34,  161-163 
reasons  for  giving,  xii 
rectangular   block    and   cylinder,   48, 

163 
square  frame  leaning  on   block,   91, 

164 
triangular  prism  and  frame,  131,  167 


Profile  lines,  21,  22 

QUATREFOIL,  101 

Railroad     track,     illustrates     vanishing 

lines,  41,  42 
Reflections,  lengthened  by  waves,  160 

length  of  vertical,  159,  160 

on  a  horizontal  surface,  156-158 

on  a  vertical  surface,  159 

when  separated  from  reflecting  sur- 
face, 158 
Rose  jar,  artistic  rendering,  17 

foot,  16,  17 

ornament  on,  17 

shoulders,  16 

study  of,  15-16 

tangential  joinings,  17 
Rocking  chair,  118 
Round  arch,  99,  103 
Round  window,  100-101 

San'  Apollinare,  Church  of,  81 
Selection  for  picture,  from  photograph,  81 ; 
see  Composition 

from  interior,  112,  115 
Shadows,  143-153 

cast  by  an  oblique  edge,  150-151 

cast   by   parallel   rays    (sun,   moon), 
144-151 

cast  by  rays  from  lamp,  143,  152 

distorted,  148-149 

in  an  interior,  152-153 

in  a  shadow-box,  144-150 

located  by  imaginary  vertical  planes, 
147 

located  by  two  points,  148 

of  a  cube,  147 

of  curves,  149 

of  natural  objects,  151-152 

on  an  obHque  surface,  151 

on  a  house,  151 

on  curved  surfaces,  149-150 

vanishing  of  light  rays,  146-147 

vanishing  of  shadow-directions,  145- 
146 
Shoulders  of  cylindrical  objects,   16;    re- 
lation to  cone  principle,  19 
Solutions  of  problems,  161-168 

cylinder  and  rectangular  block,  163 

cylinder,  cone  and  ball,  161 

square  frame  leaning  on  rectangular 
block,  164 


174 


INDEX 


triangular  prism  and  frame,  167 
Spherical  picture  plane,  141-142 
Spout,  of  cream  jug,  22 
Square  frame,  85-87 

test  of,  87 

application  of  study,  87 

leaning    on    rectangular    block, 
164 


91, 


Table  line,  explained,  3 

high  enough  on  paper,  43 

position  with  plate  on  edge,  30 

significance  in  composition,  13 

subordination  of,  44 
Taking  direction  of  vanishing  edges  with 

pencil,  55 
Teapot,  cover,  28;  ears,  28 

knob,  28 

study,  26-28 
Tests,  by  blackboard  for  vanishing  lines, 
57 

by  eye,  for  vanishing  lines,  56 

cylindrical  objects  not  vertical,  95 

hexagonal  plinth,  124 

square  frame,  87 


the  ellipse,  9 

the  eye,  a  final  test,  xii,  142 
Thumb-nail  sketches,  interiors,  115 

still-life  objects,  26,  68 
Tiled  floor,  109,  126 
Time  study,  glass  bowl,  24,  25 
Triangular  prism  and  frame,  131,  167 
Two  books,  at  difl^erent  angles,  61-62 

with  cylindrical  object,  67-68 

Vanishing  lines,  "  converging,"  37-38 

example  of,  railroad,  41-42 

oblique,    73;    see   Oblique    vanishing 
lines 

taking  direction  of,  with  pencil,  55 

tests  of,  56,  57 
Vanishing  of  parallel  planes,   52,   73-74, 

162,  note 
Vanishing  points,  abbreviation  of,  44 

numbering,  50 

oblique,    73;    see   Oblique    vanishing 
lines 

use  without  marking,  56,  80,  124 
Vanishing  traces,  73,  151 
Vertical  lines  drawn  vertical,  60 
Vignetting,  116 


175 


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